Solve Using the Quadratic Formula 16(7-3x)^2+13=17

$16 {(7 - 3 x)}^{2} + 13 = 17$

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

Subtract $17$ from both sides of the equation.

$16 {(7 - 3 x)}^{2} + 13 - 17 = 0$

Simplify $16 {(7 - 3 x)}^{2} + 13 - 17$ .

Simplify each term.

Rewrite ${(7 - 3 x)}^{2}$ as $(7 - 3 x) (7 - 3 x)$ .

$16 ((7 - 3 x) (7 - 3 x)) + 13 - 17 = 0$

Expand $(7 - 3 x) (7 - 3 x)$ using the FOIL Method.

Apply the distributive property.

$16 (7 (7 - 3 x) - 3 x (7 - 3 x)) + 13 - 17 = 0$

Apply the distributive property.

$16 (7 \cdot 7 + 7 (- 3 x) - 3 x (7 - 3 x)) + 13 - 17 = 0$

Apply the distributive property.

$16 (7 \cdot 7 + 7 (- 3 x) - 3 x \cdot 7 - 3 x (- 3 x)) + 13 - 17 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $7$ .

$16 (49 + 7 (- 3 x) - 3 x \cdot 7 - 3 x (- 3 x)) + 13 - 17 = 0$

Multiply $- 3$ by $7$ .

$16 (49 - 21 x - 3 x \cdot 7 - 3 x (- 3 x)) + 13 - 17 = 0$

Multiply $7$ by $- 3$ .

$16 (49 - 21 x - 21 x - 3 x (- 3 x)) + 13 - 17 = 0$

Rewrite using the commutative property of multiplication.

$16 (49 - 21 x - 21 x - 3 \cdot - 3 x \cdot x) + 13 - 17 = 0$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$16 (49 - 21 x - 21 x - 3 \cdot - 3 (x \cdot x)) + 13 - 17 = 0$

Multiply $x$ by $x$ .

$16 (49 - 21 x - 21 x - 3 \cdot - 3 x^{2}) + 13 - 17 = 0$

Multiply $- 3$ by $- 3$ .

$16 (49 - 21 x - 21 x + 9 x^{2}) + 13 - 17 = 0$

Subtract $21 x$ from $- 21 x$ .

$16 (49 - 42 x + 9 x^{2}) + 13 - 17 = 0$

Apply the distributive property.

$16 \cdot 49 + 16 (- 42 x) + 16 (9 x^{2}) + 13 - 17 = 0$

Simplify.

Multiply $16$ by $49$ .

$784 + 16 (- 42 x) + 16 (9 x^{2}) + 13 - 17 = 0$

Multiply $- 42$ by $16$ .

$784 - 672 x + 16 (9 x^{2}) + 13 - 17 = 0$

Multiply $9$ by $16$ .

$784 - 672 x + 144 x^{2} + 13 - 17 = 0$

Add $784$ and $13$ .

$- 672 x + 144 x^{2} + 797 - 17 = 0$

Subtract $17$ from $797$ .

$- 672 x + 144 x^{2} + 780 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 144$ , $b = - 672$ , and $c = 780$ into the quadratic formula and solve for $x$ .

$\frac{672 \pm \sqrt{{(- 672)}^{2} - 4 \cdot (144 \cdot 780)}}{2 \cdot 144}$

Simplify.

Simplify the numerator.

Raise $- 672$ to the power of $2$ .

$x = \frac{672 \pm \sqrt{451584 - 4 \cdot 144 \cdot 780}}{2 \cdot 144}$

Multiply $- 4$ by $144$ .

$x = \frac{672 \pm \sqrt{451584 - 576 \cdot 780}}{2 \cdot 144}$

Multiply $- 576$ by $780$ .

$x = \frac{672 \pm \sqrt{451584 - 449280}}{2 \cdot 144}$

Subtract $449280$ from $451584$ .

$x = \frac{672 \pm \sqrt{2304}}{2 \cdot 144}$

Rewrite $2304$ as $48^{2}$ .

$x = \frac{672 \pm \sqrt{48^{2}}}{2 \cdot 144}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{672 \pm 48}{2 \cdot 144}$

Multiply $2$ by $144$ .

$x = \frac{672 \pm 48}{288}$

Simplify $\frac{672 \pm 48}{288}$ .

$x = \frac{14 \pm 1}{6}$

The final answer is the combination of both solutions.

$x = \frac{5}{2}, \frac{13}{6}$

Solve Using the Quadratic Formula 16(7-3x)^2+13=17

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