Precalculus Examples

$625 x^{2} + 25 y^{2} = 15625$ , $y = x^{2} - 25$

Step 1

Replace all occurrences of $y$ with $x^{2} - 25$ in each equation.

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Step 1.1

Replace all occurrences of $y$ in $625 x^{2} + 25 y^{2} = 15625$ with $x^{2} - 25$ .

$625 x^{2} + 25 {(x^{2} - 25)}^{2} = 15625$

$y = x^{2} - 25$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $625 x^{2} + 25 {(x^{2} - 25)}^{2}$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(x^{2} - 25)}^{2}$ as $(x^{2} - 25) (x^{2} - 25)$ .

$625 x^{2} + 25 ((x^{2} - 25) (x^{2} - 25)) = 15625$

$y = x^{2} - 25$

Step 1.2.1.1.2

Expand $(x^{2} - 25) (x^{2} - 25)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$625 x^{2} + 25 (x^{2} (x^{2} - 25) - 25 (x^{2} - 25)) = 15625$

$y = x^{2} - 25$

Step 1.2.1.1.2.2

Apply the distributive property.

$625 x^{2} + 25 (x^{2} x^{2} + x^{2} \cdot - 25 - 25 (x^{2} - 25)) = 15625$

$y = x^{2} - 25$

Step 1.2.1.1.2.3

Apply the distributive property.

$625 x^{2} + 25 (x^{2} x^{2} + x^{2} \cdot - 25 - 25 x^{2} - 25 \cdot - 25) = 15625$

$y = x^{2} - 25$

$625 x^{2} + 25 (x^{2} x^{2} + x^{2} \cdot - 25 - 25 x^{2} - 25 \cdot - 25) = 15625$

$y = x^{2} - 25$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.2.1.1.3.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$625 x^{2} + 25 (x^{2 + 2} + x^{2} \cdot - 25 - 25 x^{2} - 25 \cdot - 25) = 15625$

$y = x^{2} - 25$

Step 1.2.1.1.3.1.1.2

Add $2$ and $2$ .

$625 x^{2} + 25 (x^{4} + x^{2} \cdot - 25 - 25 x^{2} - 25 \cdot - 25) = 15625$

$y = x^{2} - 25$

$625 x^{2} + 25 (x^{4} + x^{2} \cdot - 25 - 25 x^{2} - 25 \cdot - 25) = 15625$

$y = x^{2} - 25$

Step 1.2.1.1.3.1.2

Move $- 25$ to the left of $x^{2}$ .

$625 x^{2} + 25 (x^{4} - 25 \cdot x^{2} - 25 x^{2} - 25 \cdot - 25) = 15625$

$y = x^{2} - 25$

Step 1.2.1.1.3.1.3

Multiply $- 25$ by $- 25$ .

$625 x^{2} + 25 (x^{4} - 25 x^{2} - 25 x^{2} + 625) = 15625$

$y = x^{2} - 25$

$625 x^{2} + 25 (x^{4} - 25 x^{2} - 25 x^{2} + 625) = 15625$

$y = x^{2} - 25$

Step 1.2.1.1.3.2

Subtract $25 x^{2}$ from $- 25 x^{2}$ .

$625 x^{2} + 25 (x^{4} - 50 x^{2} + 625) = 15625$

$y = x^{2} - 25$

$625 x^{2} + 25 (x^{4} - 50 x^{2} + 625) = 15625$

$y = x^{2} - 25$

Step 1.2.1.1.4

Apply the distributive property.

$625 x^{2} + 25 x^{4} + 25 (- 50 x^{2}) + 25 \cdot 625 = 15625$

$y = x^{2} - 25$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Multiply $- 50$ by $25$ .

$625 x^{2} + 25 x^{4} - 1250 x^{2} + 25 \cdot 625 = 15625$

$y = x^{2} - 25$

Step 1.2.1.1.5.2

Multiply $25$ by $625$ .

$625 x^{2} + 25 x^{4} - 1250 x^{2} + 15625 = 15625$

$y = x^{2} - 25$

$625 x^{2} + 25 x^{4} - 1250 x^{2} + 15625 = 15625$

$y = x^{2} - 25$

$625 x^{2} + 25 x^{4} - 1250 x^{2} + 15625 = 15625$

$y = x^{2} - 25$

Step 1.2.1.2

Subtract $1250 x^{2}$ from $625 x^{2}$ .

$25 x^{4} - 625 x^{2} + 15625 = 15625$

$y = x^{2} - 25$

$25 x^{4} - 625 x^{2} + 15625 = 15625$

$y = x^{2} - 25$

$25 x^{4} - 625 x^{2} + 15625 = 15625$

$y = x^{2} - 25$

$25 x^{4} - 625 x^{2} + 15625 = 15625$

$y = x^{2} - 25$

Step 2

Solve for $x$ in $25 x^{4} - 625 x^{2} + 15625 = 15625$ .

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Step 2.1

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$25 u^{2} - 625 u + 15625 = 15625$

$u = x^{2}$

$y = x^{2} - 25$

Step 2.2

Subtract $15625$ from both sides of the equation.

$25 u^{2} - 625 u + 15625 - 15625 = 0$

$y = x^{2} - 25$

Step 2.3

Combine the opposite terms in $25 u^{2} - 625 u + 15625 - 15625$ .

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Step 2.3.1

Subtract $15625$ from $15625$ .

$25 u^{2} - 625 u + 0 = 0$

$y = x^{2} - 25$

Step 2.3.2

Add $25 u^{2} - 625 u$ and $0$ .

$25 u^{2} - 625 u = 0$

$y = x^{2} - 25$

$25 u^{2} - 625 u = 0$

$y = x^{2} - 25$

Step 2.4

Factor $25 u$ out of $25 u^{2} - 625 u$ .

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Step 2.4.1

Factor $25 u$ out of $25 u^{2}$ .

$25 u (u) - 625 u = 0$

$y = x^{2} - 25$

Step 2.4.2

Factor $25 u$ out of $- 625 u$ .

$25 u (u) + 25 u (- 25) = 0$

$y = x^{2} - 25$

Step 2.4.3

Factor $25 u$ out of $25 u (u) + 25 u (- 25)$ .

$25 u (u - 25) = 0$

$y = x^{2} - 25$

$25 u (u - 25) = 0$

$y = x^{2} - 25$

Step 2.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u = 0$

$u - 25 = 0$

$y = x^{2} - 25$

Step 2.6

Set $u$ equal to $0$ .

$u = 0$

$y = x^{2} - 25$

Step 2.7

Set $u - 25$ equal to $0$ and solve for $u$ .

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Step 2.7.1

Set $u - 25$ equal to $0$ .

$u - 25 = 0$

$y = x^{2} - 25$

Step 2.7.2

Add $25$ to both sides of the equation.

$u = 25$

$y = x^{2} - 25$

$u = 25$

$y = x^{2} - 25$

Step 2.8

The final solution is all the values that make $25 u (u - 25) = 0$ true.

$u = 0, 25$

$y = x^{2} - 25$

Step 2.9

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 0$

$x^{2} = 25$

$y = x^{2} - 25$

Step 2.10

Solve the first equation for $x$ .

$x^{2} = 0$

$y = x^{2} - 25$

Step 2.11

Solve the equation for $x$ .

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Step 2.11.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

$y = x^{2} - 25$

Step 2.11.2

Simplify $\pm \sqrt{0}$ .

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Step 2.11.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

$y = x^{2} - 25$

Step 2.11.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

$y = x^{2} - 25$

Step 2.11.2.3

Plus or minus $0$ is $0$ .

$x = 0$

$y = x^{2} - 25$

$x = 0$

$y = x^{2} - 25$

$x = 0$

$y = x^{2} - 25$

Step 2.12

Solve the second equation for $x$ .

$x^{2} = 25$

$y = x^{2} - 25$

Step 2.13

Solve the equation for $x$ .

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Step 2.13.1

Remove parentheses.

$x^{2} = 25$

$y = x^{2} - 25$

Step 2.13.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{25}$

$y = x^{2} - 25$

Step 2.13.3

Simplify $\pm \sqrt{25}$ .

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Step 2.13.3.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2}}$

$y = x^{2} - 25$

Step 2.13.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 5$

$y = x^{2} - 25$

$x = \pm 5$

$y = x^{2} - 25$

Step 2.13.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.13.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 5$

$y = x^{2} - 25$

Step 2.13.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 5$

$y = x^{2} - 25$

Step 2.13.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 5, - 5$

$y = x^{2} - 25$

$x = 5, - 5$

$y = x^{2} - 25$

$x = 5, - 5$

$y = x^{2} - 25$

Step 2.14

The solution to $25 x^{4} - 625 x^{2} + 15625 = 15625$ is $x = 0, 5, - 5$ .

$x = 0, 5, - 5$

$y = x^{2} - 25$

$x = 0, 5, - 5$

$y = x^{2} - 25$

Step 3

Replace all occurrences of $x$ with $0$ in each equation.

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Step 3.1

Replace all occurrences of $x$ in $y = x^{2} - 25$ with $0$ .

$y = {(0)}^{2} - 25$

$x = 0$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify ${(0)}^{2} - 25$ .

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Step 3.2.1.1

Raising $0$ to any positive power yields $0$ .

$y = 0 - 25$

$x = 0$

Step 3.2.1.2

Subtract $25$ from $0$ .

$y = - 25$

$x = 0$

$y = - 25$

$x = 0$

$y = - 25$

$x = 0$

$y = - 25$

$x = 0$

Step 4

Replace all occurrences of $x$ with $5$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = x^{2} - 25$ with $5$ .

$y = {(5)}^{2} - 25$

$x = 5$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify ${(5)}^{2} - 25$ .

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Step 4.2.1.1

Raise $5$ to the power of $2$ .

$y = 25 - 25$

$x = 5$

Step 4.2.1.2

Subtract $25$ from $25$ .

$y = 0$

$x = 5$

$y = 0$

$x = 5$

$y = 0$

$x = 5$

$y = 0$

$x = 5$

Step 5

Replace all occurrences of $x$ with $0$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = x^{2} - 25$ with $0$ .

$y = {(0)}^{2} - 25$

$x = 0$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify ${(0)}^{2} - 25$ .

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Step 5.2.1.1

Raising $0$ to any positive power yields $0$ .

$y = 0 - 25$

$x = 0$

Step 5.2.1.2

Subtract $25$ from $0$ .

$y = - 25$

$x = 0$

$y = - 25$

$x = 0$

$y = - 25$

$x = 0$

$y = - 25$

$x = 0$

Step 6

Replace all occurrences of $x$ with $5$ in each equation.

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Step 6.1

Replace all occurrences of $x$ in $y = x^{2} - 25$ with $5$ .

$y = {(5)}^{2} - 25$

$x = 5$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify ${(5)}^{2} - 25$ .

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Step 6.2.1.1

Raise $5$ to the power of $2$ .

$y = 25 - 25$

$x = 5$

Step 6.2.1.2

Subtract $25$ from $25$ .

$y = 0$

$x = 5$

$y = 0$

$x = 5$

$y = 0$

$x = 5$

$y = 0$

$x = 5$

Step 7

Replace all occurrences of $x$ with $- 5$ in each equation.

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Step 7.1

Replace all occurrences of $x$ in $y = x^{2} - 25$ with $- 5$ .

$y = {(- 5)}^{2} - 25$

$x = - 5$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify ${(- 5)}^{2} - 25$ .

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Step 7.2.1.1

Raise $- 5$ to the power of $2$ .

$y = 25 - 25$

$x = - 5$

Step 7.2.1.2

Subtract $25$ from $25$ .

$y = 0$

$x = - 5$

$y = 0$

$x = - 5$

$y = 0$

$x = - 5$

$y = 0$

$x = - 5$

Step 8

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, - 25)$

$(5, 0)$

$(- 5, 0)$

Step 9

The result can be shown in multiple forms.

Point Form:

$(0, - 25), (5, 0), (- 5, 0)$

Equation Form:

$x = 0, y = - 25$

$x = 5, y = 0$

$x = - 5, y = 0$

Step 10