Solve for x x+9=4(x-1)^2

$x + 9 = 4 {(x - 1)}^{2}$

Step 1

Simplify $4 {(x - 1)}^{2}$ .

Rewrite.

$x + 9 = 0 + 0 + 4 {(x - 1)}^{2}$

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$x + 9 = 4 ((x - 1) (x - 1))$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$x + 9 = 4 (x (x - 1) - 1 (x - 1))$

Apply the distributive property.

$x + 9 = 4 (x \cdot x + x \cdot - 1 - 1 (x - 1))$

Apply the distributive property.

$x + 9 = 4 (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x + 9 = 4 (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Move $- 1$ to the left of $x$ .

$x + 9 = 4 (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)$

Rewrite $- 1 x$ as $- x$ .

$x + 9 = 4 (x^{2} - x - 1 x - 1 \cdot - 1)$

Rewrite $- 1 x$ as $- x$ .

$x + 9 = 4 (x^{2} - x - x - 1 \cdot - 1)$

Multiply $- 1$ by $- 1$ .

$x + 9 = 4 (x^{2} - x - x + 1)$

Subtract $x$ from $- x$ .

$x + 9 = 4 (x^{2} - 2 x + 1)$

Apply the distributive property.

$x + 9 = 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

Simplify.

Multiply $- 2$ by $4$ .

$x + 9 = 4 x^{2} - 8 x + 4 \cdot 1$

Multiply $4$ by $1$ .

$x + 9 = 4 x^{2} - 8 x + 4$

Step 2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 x^{2} - 8 x + 4 = x + 9$

Step 3

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$4 x^{2} - 8 x + 4 - x = 9$

Subtract $x$ from $- 8 x$ .

$4 x^{2} - 9 x + 4 = 9$

Step 4

Move all terms to the left side of the equation and simplify.

Subtract $9$ from both sides of the equation.

$4 x^{2} - 9 x + 4 - 9 = 0$

Subtract $9$ from $4$ .

$4 x^{2} - 9 x - 5 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 4$ , $b = - 9$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (4 \cdot - 5)}}{2 \cdot 4}$

Step 7

Simplify.

Simplify the numerator.

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 4 \cdot - 5}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot - 5$ .

Multiply $- 4$ by $4$ .

$x = \frac{9 \pm \sqrt{81 - 16 \cdot - 5}}{2 \cdot 4}$

Multiply $- 16$ by $- 5$ .

$x = \frac{9 \pm \sqrt{81 + 80}}{2 \cdot 4}$

Add $81$ and $80$ .

$x = \frac{9 \pm \sqrt{161}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{9 \pm \sqrt{161}}{8}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{9 + \sqrt{161}}{8}, \frac{9 - \sqrt{161}}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9 + \sqrt{161}}{8}, \frac{9 - \sqrt{161}}{8}$

Decimal Form:

$x = 2.71107219 \dots, - 0.46107219 \dots$

Solve for x x+9=4(x-1)^2

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