Solve Using the Square Root Property 16x^2-48x+36=49

$16 x^{2} - 48 x + 36 = 49$

Step 1

Subtract $49$ from both sides of the equation.

$16 x^{2} - 48 x + 36 - 49 = 0$

Step 2

Subtract $49$ from $36$ .

$16 x^{2} - 48 x - 13 = 0$

Step 3

Factor by grouping.

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 $a \cdot c = 16 \cdot - 13 = - 208$ and whose sum is $b = - 48$ .

Factor $- 48$ out of $- 48 x$ .

$16 x^{2} - 48 x - 13 = 0$

Rewrite $- 48$ as $4$ plus $- 52$

$16 x^{2} + (4 - 52) x - 13 = 0$

Apply the distributive property.

$16 x^{2} + 4 x - 52 x - 13 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(16 x^{2} + 4 x) - 52 x - 13 = 0$

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

$4 x (4 x + 1) - 13 (4 x + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$(4 x + 1) (4 x - 13) = 0$

Step 4

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

$4 x + 1 = 0$

$4 x - 13 = 0$

Step 5

Set $4 x + 1$ equal to $0$ and solve for $x$ .

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Solve $4 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Divide each term in $4 x = - 1$ by $4$ and simplify.

Divide each term in $4 x = - 1$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 6

Set $4 x - 13$ equal to $0$ and solve for $x$ .

Set $4 x - 13$ equal to $0$ .

$4 x - 13 = 0$

Solve $4 x - 13 = 0$ for $x$ .

Add $13$ to both sides of the equation.

$4 x = 13$

Divide each term in $4 x = 13$ by $4$ and simplify.

Divide each term in $4 x = 13$ by $4$ .

$\frac{4 x}{4} = \frac{13}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{13}{4}$

Divide $x$ by $1$ .

$x = \frac{13}{4}$

Step 7

The final solution is all the values that make $(4 x + 1) (4 x - 13) = 0$ true.

$x = - \frac{1}{4}, \frac{13}{4}$

Solve Using the Square Root Property 16x^2-48x+36=49

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