Algebra Examples

$y = - 4 x^{2} - x + 3$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - 4 x^{2} - x + 3$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 4 x^{2} - x + 3 = 0$ .

$- 4 x^{2} - x + 3 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $- 1$ out of $- 4 x^{2} - x + 3$ .

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Step 1.2.2.1.1

Factor $- 1$ out of $- 4 x^{2}$ .

$- (4 x^{2}) - x + 3 = 0$

Step 1.2.2.1.2

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

$- (4 x^{2}) - (x) + 3 = 0$

Step 1.2.2.1.3

Rewrite $3$ as $- 1 (- 3)$ .

$- (4 x^{2}) - (x) - 1 \cdot - 3 = 0$

Step 1.2.2.1.4

Factor $- 1$ out of $- (4 x^{2}) - (x)$ .

$- (4 x^{2} + x) - 1 \cdot - 3 = 0$

Step 1.2.2.1.5

Factor $- 1$ out of $- (4 x^{2} + x) - 1 (- 3)$ .

$- (4 x^{2} + x - 3) = 0$

Step 1.2.2.2

Factor.

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Step 1.2.2.2.1

Factor by grouping.

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Step 1.2.2.2.1.1

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 $a \cdot c = 4 \cdot - 3 = - 12$ and whose sum is $b = 1$ .

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Step 1.2.2.2.1.1.1

Multiply by $1$ .

$- (4 x^{2} + 1 x - 3) = 0$

Step 1.2.2.2.1.1.2

Rewrite $1$ as $- 3$ plus $4$

$- (4 x^{2} + (- 3 + 4) x - 3) = 0$

Step 1.2.2.2.1.1.3

Apply the distributive property.

$- (4 x^{2} - 3 x + 4 x - 3) = 0$

Step 1.2.2.2.1.2

Factor out the greatest common factor from each group.

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Step 1.2.2.2.1.2.1

Group the first two terms and the last two terms.

$- ((4 x^{2} - 3 x) + 4 x - 3) = 0$

Step 1.2.2.2.1.2.2

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

$- (x (4 x - 3) + 1 (4 x - 3)) = 0$

Step 1.2.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 3$ .

$- ((4 x - 3) (x + 1)) = 0$

Step 1.2.2.2.2

Remove unnecessary parentheses.

$- (4 x - 3) (x + 1) = 0$

Step 1.2.3

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

$4 x - 3 = 0$

$x + 1 = 0$

Step 1.2.4

Set $4 x - 3$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $4 x - 3$ equal to $0$ .

$4 x - 3 = 0$

Step 1.2.4.2

Solve $4 x - 3 = 0$ for $x$ .

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Step 1.2.4.2.1

Add $3$ to both sides of the equation.

$4 x = 3$

Step 1.2.4.2.2

Divide each term in $4 x = 3$ by $4$ and simplify.

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Step 1.2.4.2.2.1

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Step 1.2.4.2.2.2

Simplify the left side.

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Step 1.2.4.2.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.4.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Step 1.2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 1.2.5

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

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Step 1.2.5.1

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

$x + 1 = 0$

Step 1.2.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.6

The final solution is all the values that make $- (4 x - 3) (x + 1) = 0$ true.

$x = \frac{3}{4}, - 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{3}{4}, 0), (- 1, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - 4 {(0)}^{2} - (0) + 3$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 4 \cdot 0^{2} - (0) + 3$

Step 2.2.2

Remove parentheses.

$y = - 4 {(0)}^{2} - (0) + 3$

Step 2.2.3

Simplify $- 4 {(0)}^{2} - (0) + 3$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

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

$y = - 4 \cdot 0 - (0) + 3$

Step 2.2.3.1.2

Multiply $- 4$ by $0$ .

$y = 0 - (0) + 3$

Step 2.2.3.1.3

Multiply $- 1$ by $0$ .

$y = 0 + 0 + 3$

Step 2.2.3.2

Simplify by adding numbers.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 + 3$

Step 2.2.3.2.2

Add $0$ and $3$ .

$y = 3$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 3)$

Step 3

List the intersections.

x-intercept(s): $(\frac{3}{4}, 0), (- 1, 0)$

y-intercept(s): $(0, 3)$

Step 4