Solve 4x^2+112x=121

$$4 { x }^{ 2 } +112x=121$$

Solve for x

$$x=\frac{\sqrt{905}}{2}-14\approx 1.041608956$$

$$x=-\frac{\sqrt{905}}{2}-14\approx -29.041608956$$

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.

$$4x^{2}+112x=121$$

Subtract 121 from both sides of the equation.

$$4x^{2}+112x-121=121-121$$

Subtracting 121 from itself leaves 0.

$$4x^{2}+112x-121=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 112 for b, and -121 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-112±\sqrt{112^{2}-4\times 4\left(-121\right)}}{2\times 4}$$

Square 112.

$$x=\frac{-112±\sqrt{12544-4\times 4\left(-121\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-112±\sqrt{12544-16\left(-121\right)}}{2\times 4}$$

Multiply -16 times -121.

$$x=\frac{-112±\sqrt{12544+1936}}{2\times 4}$$

Add 12544 to 1936.

$$x=\frac{-112±\sqrt{14480}}{2\times 4}$$

Take the square root of 14480.

$$x=\frac{-112±4\sqrt{905}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-112±4\sqrt{905}}{8}$$

Now solve the equation x=\frac{-112±4\sqrt{905}}{8} when ± is plus. Add -112 to 4\sqrt{905}.

$$x=\frac{4\sqrt{905}-112}{8}$$

Divide -112+4\sqrt{905} by 8.

$$x=\frac{\sqrt{905}}{2}-14$$

Now solve the equation x=\frac{-112±4\sqrt{905}}{8} when ± is minus. Subtract 4\sqrt{905} from -112.

$$x=\frac{-4\sqrt{905}-112}{8}$$

Divide -112-4\sqrt{905} by 8.

$$x=-\frac{\sqrt{905}}{2}-14$$

The equation is now solved.

$$x=\frac{\sqrt{905}}{2}-14 x=-\frac{\sqrt{905}}{2}-14$$

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.

$$4x^{2}+112x=121$$

Divide both sides by 4.

$$\frac{4x^{2}+112x}{4}=\frac{121}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{112}{4}x=\frac{121}{4}$$

Divide 112 by 4.

$$x^{2}+28x=\frac{121}{4}$$

Divide 28, the coefficient of the x term, by 2 to get 14. Then add the square of 14 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+28x+14^{2}=\frac{121}{4}+14^{2}$$

Square 14.

$$x^{2}+28x+196=\frac{121}{4}+196$$

Add \frac{121}{4} to 196.

$$x^{2}+28x+196=\frac{905}{4}$$

Factor x^{2}+28x+196. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+14\right)^{2}=\frac{905}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+14\right)^{2}}=\sqrt{\frac{905}{4}}$$

Simplify.

$$x+14=\frac{\sqrt{905}}{2} x+14=-\frac{\sqrt{905}}{2}$$

Subtract 14 from both sides of the equation.

$$x=\frac{\sqrt{905}}{2}-14 x=-\frac{\sqrt{905}}{2}-14$$