Solve 1+x^2-20x+24=0

$$1 + x ^ { 2 } - 20 x + 24 = 0$$

Solve for x

$$x=5\sqrt{3}+10\approx 18.660254038$$

$$x=10-5\sqrt{3}\approx 1.339745962$$

Steps Using the Quadratic Formula

Add 1 and 24 to get 25.

$$25+x^{2}-20x=0$$

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.

$$x^{2}-20x+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -20 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 25}}{2}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 25}}{2}$$

Multiply -4 times 25.

$$x=\frac{-\left(-20\right)±\sqrt{400-100}}{2}$$

Add 400 to -100.

$$x=\frac{-\left(-20\right)±\sqrt{300}}{2}$$

Take the square root of 300.

$$x=\frac{-\left(-20\right)±10\sqrt{3}}{2}$$

The opposite of -20 is 20.

$$x=\frac{20±10\sqrt{3}}{2}$$

Now solve the equation x=\frac{20±10\sqrt{3}}{2} when ± is plus. Add 20 to 10\sqrt{3}.

$$x=\frac{10\sqrt{3}+20}{2}$$

Divide 20+10\sqrt{3} by 2.

$$x=5\sqrt{3}+10$$

Now solve the equation x=\frac{20±10\sqrt{3}}{2} when ± is minus. Subtract 10\sqrt{3} from 20.

$$x=\frac{20-10\sqrt{3}}{2}$$

Divide 20-10\sqrt{3} by 2.

$$x=10-5\sqrt{3}$$

The equation is now solved.

$$x=5\sqrt{3}+10 x=10-5\sqrt{3}$$

Steps for Completing the Square

Add 1 and 24 to get 25.

$$25+x^{2}-20x=0$$

Subtract 25 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-20x=-25$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-20x+\left(-10\right)^{2}=-25+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=-25+100$$

Add -25 to 100.

$$x^{2}-20x+100=75$$

Factor x^{2}-20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-10\right)^{2}=75$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{75}$$

Simplify.

$$x-10=5\sqrt{3} x-10=-5\sqrt{3}$$

Add 10 to both sides of the equation.

$$x=5\sqrt{3}+10 x=10-5\sqrt{3}$$