Solve Using the Quadratic Formula 3(x+7)^2+17=59

$3 {(x + 7)}^{2} + 17 = 59$

Step 1

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

Subtract $59$ from both sides of the equation.

$3 {(x + 7)}^{2} + 17 - 59 = 0$

Simplify $3 {(x + 7)}^{2} + 17 - 59$ .

Simplify each term.

Rewrite ${(x + 7)}^{2}$ as $(x + 7) (x + 7)$ .

$3 ((x + 7) (x + 7)) + 17 - 59 = 0$

Expand $(x + 7) (x + 7)$ using the FOIL Method.

Apply the distributive property.

$3 (x (x + 7) + 7 (x + 7)) + 17 - 59 = 0$

Apply the distributive property.

$3 (x \cdot x + x \cdot 7 + 7 (x + 7)) + 17 - 59 = 0$

Apply the distributive property.

$3 (x \cdot x + x \cdot 7 + 7 x + 7 \cdot 7) + 17 - 59 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$3 (x^{2} + x \cdot 7 + 7 x + 7 \cdot 7) + 17 - 59 = 0$

Move $7$ to the left of $x$ .

$3 (x^{2} + 7 \cdot x + 7 x + 7 \cdot 7) + 17 - 59 = 0$

Multiply $7$ by $7$ .

$3 (x^{2} + 7 x + 7 x + 49) + 17 - 59 = 0$

Add $7 x$ and $7 x$ .

$3 (x^{2} + 14 x + 49) + 17 - 59 = 0$

Apply the distributive property.

$3 x^{2} + 3 (14 x) + 3 \cdot 49 + 17 - 59 = 0$

Simplify.

Multiply $14$ by $3$ .

$3 x^{2} + 42 x + 3 \cdot 49 + 17 - 59 = 0$

Multiply $3$ by $49$ .

$3 x^{2} + 42 x + 147 + 17 - 59 = 0$

Add $147$ and $17$ .

$3 x^{2} + 42 x + 164 - 59 = 0$

Subtract $59$ from $164$ .

$3 x^{2} + 42 x + 105 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 3$ , $b = 42$ , and $c = 105$ into the quadratic formula and solve for $x$ .

$\frac{- 42 \pm \sqrt{42^{2} - 4 \cdot (3 \cdot 105)}}{2 \cdot 3}$

Step 4

Simplify.

Simplify the numerator.

Raise $42$ to the power of $2$ .

$x = \frac{- 42 \pm \sqrt{1764 - 4 \cdot 3 \cdot 105}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 105$ .

Multiply $- 4$ by $3$ .

$x = \frac{- 42 \pm \sqrt{1764 - 12 \cdot 105}}{2 \cdot 3}$

Multiply $- 12$ by $105$ .

$x = \frac{- 42 \pm \sqrt{1764 - 1260}}{2 \cdot 3}$

Subtract $1260$ from $1764$ .

$x = \frac{- 42 \pm \sqrt{504}}{2 \cdot 3}$

Rewrite $504$ as $6^{2} \cdot 14$ .

Factor $36$ out of $504$ .

$x = \frac{- 42 \pm \sqrt{36 (14)}}{2 \cdot 3}$

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 42 \pm \sqrt{6^{2} \cdot 14}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{- 42 \pm 6 \sqrt{14}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 42 \pm 6 \sqrt{14}}{6}$

Simplify $\frac{- 42 \pm 6 \sqrt{14}}{6}$ .

$x = - 7 \pm \sqrt{14}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - 7 \pm \sqrt{14}$

Decimal Form:

$x = - 3.25834261 \dots, - 10.74165738 \dots$

Solve Using the Quadratic Formula 3(x+7)^2+17=59

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