Algebra Examples

$y = \frac{1}{4} x^{2} + x - 6$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

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

$0 = \frac{1}{4} x^{2} + x - 6$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $\frac{1}{4} x^{2} + x - 6 = 0$ .

$\frac{1}{4} x^{2} + x - 6 = 0$

Step 1.2.2

Combine $\frac{1}{4}$ and $x^{2}$ .

$\frac{x^{2}}{4} + x - 6 = 0$

Step 1.2.3

Multiply through by the least common denominator $4$ , then simplify.

Tap for more steps...

Step 1.2.3.1

Apply the distributive property.

$4 (\frac{x^{2}}{4}) + 4 x + 4 \cdot - 6 = 0$

Step 1.2.3.2

Simplify.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.3.2.1.1

Cancel the common factor.

$4 (\frac{x^{2}}{4}) + 4 x + 4 \cdot - 6 = 0$

Step 1.2.3.2.1.2

Rewrite the expression.

$x^{2} + 4 x + 4 \cdot - 6 = 0$

Step 1.2.3.2.2

Multiply $4$ by $- 6$ .

$x^{2} + 4 x - 24 = 0$

Step 1.2.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.5

Substitute the values $a = 1$ , $b = 4$ , and $c = - 24$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot - 24)}}{2 \cdot 1}$

Step 1.2.6

Simplify.

Tap for more steps...

Step 1.2.6.1

Simplify the numerator.

Tap for more steps...

Step 1.2.6.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 24}}{2 \cdot 1}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 1 \cdot - 24$ .

Tap for more steps...

Step 1.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 24}}{2 \cdot 1}$

Step 1.2.6.1.2.2

Multiply $- 4$ by $- 24$ .

$x = \frac{- 4 \pm \sqrt{16 + 96}}{2 \cdot 1}$

Step 1.2.6.1.3

Add $16$ and $96$ .

$x = \frac{- 4 \pm \sqrt{112}}{2 \cdot 1}$

Step 1.2.6.1.4

Rewrite $112$ as $4^{2} \cdot 7$ .

Tap for more steps...

Step 1.2.6.1.4.1

Factor $16$ out of $112$ .

$x = \frac{- 4 \pm \sqrt{16 (7)}}{2 \cdot 1}$

Step 1.2.6.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 1}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 4 \sqrt{7}}{2 \cdot 1}$

Step 1.2.6.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 \sqrt{7}}{2}$

Step 1.2.6.3

Simplify $\frac{- 4 \pm 4 \sqrt{7}}{2}$ .

$x = - 2 \pm 2 \sqrt{7}$

Step 1.2.7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 1.2.7.1

Simplify the numerator.

Tap for more steps...

Step 1.2.7.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 24}}{2 \cdot 1}$

Step 1.2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 24$ .

Tap for more steps...

Step 1.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 24}}{2 \cdot 1}$

Step 1.2.7.1.2.2

Multiply $- 4$ by $- 24$ .

$x = \frac{- 4 \pm \sqrt{16 + 96}}{2 \cdot 1}$

Step 1.2.7.1.3

Add $16$ and $96$ .

$x = \frac{- 4 \pm \sqrt{112}}{2 \cdot 1}$

Step 1.2.7.1.4

Rewrite $112$ as $4^{2} \cdot 7$ .

Tap for more steps...

Step 1.2.7.1.4.1

Factor $16$ out of $112$ .

$x = \frac{- 4 \pm \sqrt{16 (7)}}{2 \cdot 1}$

Step 1.2.7.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 1}$

Step 1.2.7.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 4 \sqrt{7}}{2 \cdot 1}$

Step 1.2.7.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 \sqrt{7}}{2}$

Step 1.2.7.3

Simplify $\frac{- 4 \pm 4 \sqrt{7}}{2}$ .

$x = - 2 \pm 2 \sqrt{7}$

Step 1.2.7.4

Change the $\pm$ to $+$ .

$x = - 2 + 2 \sqrt{7}$

Step 1.2.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 1.2.8.1

Simplify the numerator.

Tap for more steps...

Step 1.2.8.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 24}}{2 \cdot 1}$

Step 1.2.8.1.2

Multiply $- 4 \cdot 1 \cdot - 24$ .

Tap for more steps...

Step 1.2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 24}}{2 \cdot 1}$

Step 1.2.8.1.2.2

Multiply $- 4$ by $- 24$ .

$x = \frac{- 4 \pm \sqrt{16 + 96}}{2 \cdot 1}$

Step 1.2.8.1.3

Add $16$ and $96$ .

$x = \frac{- 4 \pm \sqrt{112}}{2 \cdot 1}$

Step 1.2.8.1.4

Rewrite $112$ as $4^{2} \cdot 7$ .

Tap for more steps...

Step 1.2.8.1.4.1

Factor $16$ out of $112$ .

$x = \frac{- 4 \pm \sqrt{16 (7)}}{2 \cdot 1}$

Step 1.2.8.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 1}$

Step 1.2.8.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 4 \sqrt{7}}{2 \cdot 1}$

Step 1.2.8.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 \sqrt{7}}{2}$

Step 1.2.8.3

Simplify $\frac{- 4 \pm 4 \sqrt{7}}{2}$ .

$x = - 2 \pm 2 \sqrt{7}$

Step 1.2.8.4

Change the $\pm$ to $-$ .

$x = - 2 - 2 \sqrt{7}$

Step 1.2.9

The final answer is the combination of both solutions.

$x = - 2 + 2 \sqrt{7}, - 2 - 2 \sqrt{7}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 2 + 2 \sqrt{7}, 0), (- 2 - 2 \sqrt{7}, 0)$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

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

$y = \frac{1}{4} \cdot {(0)}^{2} + 0 - 6$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = \frac{1}{4} \cdot {(0)}^{2} + 0 - 6$

Step 2.2.2

Remove parentheses.

$y = \frac{1}{4} \cdot 0^{2} + 0 - 6$

Step 2.2.3

Remove parentheses.

$y = \frac{1}{4} \cdot {(0)}^{2} + 0 - 6$

Step 2.2.4

Simplify $\frac{1}{4} \cdot {(0)}^{2} + 0 - 6$ .

Tap for more steps...

Step 2.2.4.1

Simplify each term.

Tap for more steps...

Step 2.2.4.1.1

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

$y = \frac{1}{4} \cdot 0 + 0 - 6$

Step 2.2.4.1.2

Multiply $\frac{1}{4}$ by $0$ .

$y = 0 + 0 - 6$

Step 2.2.4.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 - 6$

Step 2.2.4.2.2

Subtract $6$ from $0$ .

$y = - 6$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 6)$

Step 3

List the intersections.

x-intercept(s): $(- 2 + 2 \sqrt{7}, 0), (- 2 - 2 \sqrt{7}, 0)$

y-intercept(s): $(0, - 6)$

Step 4