Precalculus Examples

$y = - 5 x^{2} + 2 x + 7$ , $y = 0$

Step 1

Eliminate the equal sides of each equation and combine.

$- 5 x^{2} + 2 x + 7 = 0$

Step 2

Solve $- 5 x^{2} + 2 x + 7 = 0$ for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $- 1$ out of $- 5 x^{2} + 2 x + 7$ .

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Step 2.1.1.1

Factor $- 1$ out of $- 5 x^{2}$ .

$- (5 x^{2}) + 2 x + 7 = 0$

Step 2.1.1.2

Factor $- 1$ out of $2 x$ .

$- (5 x^{2}) - (- 2 x) + 7 = 0$

Step 2.1.1.3

Rewrite $7$ as $- 1 (- 7)$ .

$- (5 x^{2}) - (- 2 x) - 1 \cdot - 7 = 0$

Step 2.1.1.4

Factor $- 1$ out of $- (5 x^{2}) - (- 2 x)$ .

$- (5 x^{2} - 2 x) - 1 \cdot - 7 = 0$

Step 2.1.1.5

Factor $- 1$ out of $- (5 x^{2} - 2 x) - 1 (- 7)$ .

$- (5 x^{2} - 2 x - 7) = 0$

Step 2.1.2

Factor.

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Step 2.1.2.1

Factor by grouping.

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Step 2.1.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 7 = - 35$ and whose sum is $b = - 2$ .

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Step 2.1.2.1.1.1

Factor $- 2$ out of $- 2 x$ .

$- (5 x^{2} - 2 x - 7) = 0$

Step 2.1.2.1.1.2

Rewrite $- 2$ as $5$ plus $- 7$

$- (5 x^{2} + (5 - 7) x - 7) = 0$

Step 2.1.2.1.1.3

Apply the distributive property.

$- (5 x^{2} + 5 x - 7 x - 7) = 0$

Step 2.1.2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1.2.1

Group the first two terms and the last two terms.

$- ((5 x^{2} + 5 x) - 7 x - 7) = 0$

Step 2.1.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (5 x (x + 1) - 7 (x + 1)) = 0$

Step 2.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$- ((x + 1) (5 x - 7)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$- (x + 1) (5 x - 7) = 0$

Step 2.2

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

$x + 1 = 0$

$5 x - 7 = 0 + y = 0$

Step 2.3

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.4

Set $5 x - 7$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $5 x - 7$ equal to $0$ .

$5 x - 7 = 0$

Step 2.4.2

Solve $5 x - 7 = 0$ for $x$ .

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Step 2.4.2.1

Add $7$ to both sides of the equation.

$5 x = 7$

Step 2.4.2.2

Divide each term in $5 x = 7$ by $5$ and simplify.

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Step 2.4.2.2.1

Divide each term in $5 x = 7$ by $5$ .

$\frac{5 x}{5} = \frac{7}{5}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{7}{5}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{5}$

Step 2.5

The final solution is all the values that make $- (x + 1) (5 x - 7) = 0$ true.

$x = - 1, \frac{7}{5}$

Step 3

Substitute $- 1$ for $x$ .

$y = 0$

Step 4

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

$(- 1, 0)$

$(\frac{7}{5}, 0)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(- 1, 0), (\frac{7}{5}, 0)$

Equation Form:

$x = - 1, y = 0$

$x = \frac{7}{5}, y = 0$

Step 6