Solve 176x=121+64x^2

$$176 x = 121 + 64 x ^ { 2 }$$

Solve for x

$$x = \frac{11}{8} = 1\frac{3}{8} = 1.375$$

Steps Using the Quadratic Formula

Subtract 121 from both sides.

$$176x-121=64x^{2}$$

Subtract 64x^{2} from both sides.

$$176x-121-64x^{2}=0$$

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.

$$-64x^{2}+176x-121=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -64 for a, 176 for b, and -121 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-176±\sqrt{176^{2}-4\left(-64\right)\left(-121\right)}}{2\left(-64\right)}$$

Square 176.

$$x=\frac{-176±\sqrt{30976-4\left(-64\right)\left(-121\right)}}{2\left(-64\right)}$$

Multiply -4 times -64.

$$x=\frac{-176±\sqrt{30976+256\left(-121\right)}}{2\left(-64\right)}$$

Multiply 256 times -121.

$$x=\frac{-176±\sqrt{30976-30976}}{2\left(-64\right)}$$

Add 30976 to -30976.

$$x=\frac{-176±\sqrt{0}}{2\left(-64\right)}$$

Take the square root of 0.

$$x=-\frac{176}{2\left(-64\right)}$$

Multiply 2 times -64.

$$x=-\frac{176}{-128}$$

Reduce the fraction \frac{-176}{-128} to lowest terms by extracting and canceling out 16.

$$x=\frac{11}{8}$$

Steps for Completing the Square

Subtract 64x^{2} from both sides.

$$176x-64x^{2}=121$$

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.

$$-64x^{2}+176x=121$$

Divide both sides by -64.

$$\frac{-64x^{2}+176x}{-64}=\frac{121}{-64}$$

Dividing by -64 undoes the multiplication by -64.

$$x^{2}+\frac{176}{-64}x=\frac{121}{-64}$$

Reduce the fraction \frac{176}{-64} to lowest terms by extracting and canceling out 16.

$$x^{2}-\frac{11}{4}x=\frac{121}{-64}$$

Divide 121 by -64.

$$x^{2}-\frac{11}{4}x=-\frac{121}{64}$$

Divide -\frac{11}{4}, the coefficient of the x term, by 2 to get -\frac{11}{8}. Then add the square of -\frac{11}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{4}x+\left(-\frac{11}{8}\right)^{2}=-\frac{121}{64}+\left(-\frac{11}{8}\right)^{2}$$

Square -\frac{11}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=\frac{-121+121}{64}$$

Add -\frac{121}{64} to \frac{121}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=0$$

Factor x^{2}-\frac{11}{4}x+\frac{121}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{8}\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{8}\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-\frac{11}{8}=0 x-\frac{11}{8}=0$$

Add \frac{11}{8} to both sides of the equation.

$$x=\frac{11}{8} x=\frac{11}{8}$$

The equation is now solved. Solutions are the same.

$$x=\frac{11}{8}$$