Solve 16x^2-64x=-48

$$16 x ^ { 2 } - 64 x = - 48$$

Solve for x

$$x=1$$

$$x=3$$

Steps Using Factoring By Grouping

Add 48 to both sides.

$$16x^{2}-64x+48=0$$

Divide both sides by 16.

$$x^{2}-4x+3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\times 3=3$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-3 b=-1$$

Rewrite x^{2}-4x+3 as \left(x^{2}-3x\right)+\left(-x+3\right).

$$\left(x^{2}-3x\right)+\left(-x+3\right)$$

Factor out x in the first and -1 in the second group.

$$x\left(x-3\right)-\left(x-3\right)$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(x-1\right)$$

To find equation solutions, solve x-3=0 and x-1=0.

$$x=3 x=1$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$16x^{2}-64x=-48$$

Add 48 to both sides of the equation.

$$16x^{2}-64x-\left(-48\right)=-48-\left(-48\right)$$

Subtracting -48 from itself leaves 0.

$$16x^{2}-64x-\left(-48\right)=0$$

Subtract -48 from 0.

$$16x^{2}-64x+48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -64 for b, and 48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\times 16\times 48}}{2\times 16}$$

Square -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\times 16\times 48}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-64\right)±\sqrt{4096-64\times 48}}{2\times 16}$$

Multiply -64 times 48.

$$x=\frac{-\left(-64\right)±\sqrt{4096-3072}}{2\times 16}$$

Add 4096 to -3072.

$$x=\frac{-\left(-64\right)±\sqrt{1024}}{2\times 16}$$

Take the square root of 1024.

$$x=\frac{-\left(-64\right)±32}{2\times 16}$$

The opposite of -64 is 64.

$$x=\frac{64±32}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{64±32}{32}$$

Now solve the equation x=\frac{64±32}{32} when ± is plus. Add 64 to 32.

$$x=\frac{96}{32}$$

Divide 96 by 32.

$$x=3$$

Now solve the equation x=\frac{64±32}{32} when ± is minus. Subtract 32 from 64.

$$x=\frac{32}{32}$$

Divide 32 by 32.

$$x=1$$

The equation is now solved.

$$x=3 x=1$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$16x^{2}-64x=-48$$

Divide both sides by 16.

$$\frac{16x^{2}-64x}{16}=-\frac{48}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{64}{16}\right)x=-\frac{48}{16}$$

Divide -64 by 16.

$$x^{2}-4x=-\frac{48}{16}$$

Divide -48 by 16.

$$x^{2}-4x=-3$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4x+\left(-2\right)^{2}=-3+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-3+4$$

Add -3 to 4.

$$x^{2}-4x+4=1$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{1}$$

Simplify.

$$x-2=1 x-2=-1$$

Add 2 to both sides of the equation.

$$x=3 x=1$$