Solve x^2+5x-7x+20=60

$${ x }^{ 2 } +5x-7x+20 = 60$$

Solve for x

$$x=\sqrt{41}+1\approx 7.403124237$$

$$x=1-\sqrt{41}\approx -5.403124237$$

Steps Using the Quadratic Formula

Combine 5x and -7x to get -2x.

$$x^{2}-2x+20=60$$

Subtract 60 from both sides.

$$x^{2}-2x+20-60=0$$

Subtract 60 from 20 to get -40.

$$x^{2}-2x-40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-40\right)}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-40\right)}}{2}$$

Multiply -4 times -40.

$$x=\frac{-\left(-2\right)±\sqrt{4+160}}{2}$$

Add 4 to 160.

$$x=\frac{-\left(-2\right)±\sqrt{164}}{2}$$

Take the square root of 164.

$$x=\frac{-\left(-2\right)±2\sqrt{41}}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{41}}{2}$$

Now solve the equation x=\frac{2±2\sqrt{41}}{2} when ± is plus. Add 2 to 2\sqrt{41}.

$$x=\frac{2\sqrt{41}+2}{2}$$

Divide 2+2\sqrt{41} by 2.

$$x=\sqrt{41}+1$$

Now solve the equation x=\frac{2±2\sqrt{41}}{2} when ± is minus. Subtract 2\sqrt{41} from 2.

$$x=\frac{2-2\sqrt{41}}{2}$$

Divide 2-2\sqrt{41} by 2.

$$x=1-\sqrt{41}$$

The equation is now solved.

$$x=\sqrt{41}+1 x=1-\sqrt{41}$$

Steps for Completing the Square

Combine 5x and -7x to get -2x.

$$x^{2}-2x+20=60$$

Subtract 20 from both sides.

$$x^{2}-2x=60-20$$

Subtract 20 from 60 to get 40.

$$x^{2}-2x=40$$

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.

$$x^{2}-2x+1=40+1$$

Add 40 to 1.

$$x^{2}-2x+1=41$$

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}.

$$\left(x-1\right)^{2}=41$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{41}$$

Simplify.

$$x-1=\sqrt{41} x-1=-\sqrt{41}$$

Add 1 to both sides of the equation.

$$x=\sqrt{41}+1 x=1-\sqrt{41}$$