Solve (x+2)^2-90=5(0,5x-17)

$$( x + 2 ) ^ { 2 } - 90 = 5 ( 0,5 x - 17 )$$

Solve for x

$$x=-2$$

$$x=0,5$$

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}.

$$x^{2}+4x+4-90=5\left(0,5x-17\right)$$

Subtract 90 from 4 to get -86.

$$x^{2}+4x-86=5\left(0,5x-17\right)$$

Use the distributive property to multiply 5 by 0,5x-17.

$$x^{2}+4x-86=2,5x-85$$

Subtract 2,5x from both sides.

$$x^{2}+4x-86-2,5x=-85$$

Combine 4x and -2,5x to get 1,5x.

$$x^{2}+1,5x-86=-85$$

Add 85 to both sides.

$$x^{2}+1,5x-86+85=0$$

Add -86 and 85 to get -1.

$$x^{2}+1,5x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1,5 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1,5±\sqrt{1,5^{2}-4\left(-1\right)}}{2}$$

Square 1,5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-1,5±\sqrt{2,25-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$x=\frac{-1,5±\sqrt{2,25+4}}{2}$$

Add 2,25 to 4.

$$x=\frac{-1,5±\sqrt{6,25}}{2}$$

Take the square root of 6,25.

$$x=\frac{-1,5±\frac{5}{2}}{2}$$

Now solve the equation x=\frac{-1,5±\frac{5}{2}}{2} when ± is plus. Add -1,5 to \frac{5}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{1}{2}$$

Now solve the equation x=\frac{-1,5±\frac{5}{2}}{2} when ± is minus. Subtract \frac{5}{2} from -1,5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide -4 by 2.

$$x=-2$$

The equation is now solved.

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

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}.

$$x^{2}+4x+4-90=5\left(0,5x-17\right)$$

Subtract 90 from 4 to get -86.

$$x^{2}+4x-86=5\left(0,5x-17\right)$$

Use the distributive property to multiply 5 by 0,5x-17.

$$x^{2}+4x-86=2,5x-85$$

Subtract 2,5x from both sides.

$$x^{2}+4x-86-2,5x=-85$$

Combine 4x and -2,5x to get 1,5x.

$$x^{2}+1,5x-86=-85$$

Add 86 to both sides.

$$x^{2}+1,5x=-85+86$$

Add -85 and 86 to get 1.

$$x^{2}+1,5x=1$$

Divide 1,5, the coefficient of the x term, by 2 to get 0,75. Then add the square of 0,75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+1,5x+0,75^{2}=1+0,75^{2}$$

Square 0,75 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+1,5x+0,5625=1+0,5625$$

Add 1 to 0,5625.

$$x^{2}+1,5x+0,5625=1,5625$$

Factor x^{2}+1,5x+0,5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+0,75\right)^{2}=1,5625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+0,75\right)^{2}}=\sqrt{1,5625}$$

Simplify.

$$x+0,75=\frac{5}{4} x+0,75=-\frac{5}{4}$$

Subtract 0,75 from both sides of the equation.

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