Solve 2=-4(x-2)^2+15

$$2 = - 4 ( x - 2 ) ^ { 2 } + 15$$

Solve for x

$$x=\frac{\sqrt{13}}{2}+2\approx 3.802775638$$

$$x=-\frac{\sqrt{13}}{2}+2\approx 0.197224362$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$2=-4\left(x^{2}-4x+4\right)+15$$

Use the distributive property to multiply -4 by x^{2}-4x+4.

$$2=-4x^{2}+16x-16+15$$

Add -16 and 15 to get -1.

$$2=-4x^{2}+16x-1$$

Swap sides so that all variable terms are on the left hand side.

$$-4x^{2}+16x-1=2$$

Subtract 2 from both sides.

$$-4x^{2}+16x-1-2=0$$

Subtract 2 from -1 to get -3.

$$-4x^{2}+16x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 16 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-4\right)\left(-3\right)}}{2\left(-4\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-4\right)\left(-3\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-16±\sqrt{256+16\left(-3\right)}}{2\left(-4\right)}$$

Multiply 16 times -3.

$$x=\frac{-16±\sqrt{256-48}}{2\left(-4\right)}$$

Add 256 to -48.

$$x=\frac{-16±\sqrt{208}}{2\left(-4\right)}$$

Take the square root of 208.

$$x=\frac{-16±4\sqrt{13}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-16±4\sqrt{13}}{-8}$$

Now solve the equation x=\frac{-16±4\sqrt{13}}{-8} when ± is plus. Add -16 to 4\sqrt{13}.

$$x=\frac{4\sqrt{13}-16}{-8}$$

Divide -16+4\sqrt{13} by -8.

$$x=-\frac{\sqrt{13}}{2}+2$$

Now solve the equation x=\frac{-16±4\sqrt{13}}{-8} when ± is minus. Subtract 4\sqrt{13} from -16.

$$x=\frac{-4\sqrt{13}-16}{-8}$$

Divide -16-4\sqrt{13} by -8.

$$x=\frac{\sqrt{13}}{2}+2$$

The equation is now solved.

$$x=-\frac{\sqrt{13}}{2}+2 x=\frac{\sqrt{13}}{2}+2$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$2=-4\left(x^{2}-4x+4\right)+15$$

Use the distributive property to multiply -4 by x^{2}-4x+4.

$$2=-4x^{2}+16x-16+15$$

Add -16 and 15 to get -1.

$$2=-4x^{2}+16x-1$$

Swap sides so that all variable terms are on the left hand side.

$$-4x^{2}+16x-1=2$$

Add 1 to both sides.

$$-4x^{2}+16x=2+1$$

Add 2 and 1 to get 3.

$$-4x^{2}+16x=3$$

Divide both sides by -4.

$$\frac{-4x^{2}+16x}{-4}=\frac{3}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{16}{-4}x=\frac{3}{-4}$$

Divide 16 by -4.

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

Divide 3 by -4.

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

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.

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

Square -2.

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

Add -\frac{3}{4} to 4.

$$x^{2}-4x+4=\frac{13}{4}$$

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}.

$$\left(x-2\right)^{2}=\frac{13}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{13}{4}}$$

Simplify.

$$x-2=\frac{\sqrt{13}}{2} x-2=-\frac{\sqrt{13}}{2}$$

Add 2 to both sides of the equation.

$$x=\frac{\sqrt{13}}{2}+2 x=-\frac{\sqrt{13}}{2}+2$$