Solve 4x+11=2x^2

$$4 x + 11 = 2 x ^ { 2 }$$

Solve for x

$$x=\frac{\sqrt{26}}{2}+1\approx 3.549509757$$

$$x=-\frac{\sqrt{26}}{2}+1\approx -1.549509757$$

Steps Using the Quadratic Formula

Subtract 2x^{2} from both sides.

$$4x+11-2x^{2}=0$$

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.

$$-2x^{2}+4x+11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 4 for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-2\right)\times 11}}{2\left(-2\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-2\right)\times 11}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-4±\sqrt{16+8\times 11}}{2\left(-2\right)}$$

Multiply 8 times 11.

$$x=\frac{-4±\sqrt{16+88}}{2\left(-2\right)}$$

Add 16 to 88.

$$x=\frac{-4±\sqrt{104}}{2\left(-2\right)}$$

Take the square root of 104.

$$x=\frac{-4±2\sqrt{26}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-4±2\sqrt{26}}{-4}$$

Now solve the equation x=\frac{-4±2\sqrt{26}}{-4} when ± is plus. Add -4 to 2\sqrt{26}.

$$x=\frac{2\sqrt{26}-4}{-4}$$

Divide -4+2\sqrt{26} by -4.

$$x=-\frac{\sqrt{26}}{2}+1$$

Now solve the equation x=\frac{-4±2\sqrt{26}}{-4} when ± is minus. Subtract 2\sqrt{26} from -4.

$$x=\frac{-2\sqrt{26}-4}{-4}$$

Divide -4-2\sqrt{26} by -4.

$$x=\frac{\sqrt{26}}{2}+1$$

The equation is now solved.

$$x=-\frac{\sqrt{26}}{2}+1 x=\frac{\sqrt{26}}{2}+1$$

Steps for Completing the Square

Subtract 2x^{2} from both sides.

$$4x+11-2x^{2}=0$$

Subtract 11 from both sides. Anything subtracted from zero gives its negation.

$$4x-2x^{2}=-11$$

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.

$$-2x^{2}+4x=-11$$

Divide both sides by -2.

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

Dividing by -2 undoes the multiplication by -2.

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

Divide 4 by -2.

$$x^{2}-2x=-\frac{11}{-2}$$

Divide -11 by -2.

$$x^{2}-2x=\frac{11}{2}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=\frac{11}{2}+1$$

Add \frac{11}{2} to 1.

$$x^{2}-2x+1=\frac{13}{2}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=\frac{13}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{13}{2}}$$

Simplify.

$$x-1=\frac{\sqrt{26}}{2} x-1=-\frac{\sqrt{26}}{2}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{26}}{2}+1 x=-\frac{\sqrt{26}}{2}+1$$