Solve 5x-10=2*10x-70+2+x^2

$$5x-10 = 2 \cdot 10x-70+2+ { x }^{ 2 }$$

Solve for x

$$x = \frac{\sqrt{457} - 15}{2} \approx 3.188779163$$

$$x=\frac{-\sqrt{457}-15}{2}\approx -18.188779163$$

Steps Using the Quadratic Formula

Multiply 2 and 10 to get 20.

$$5x-10=20x-70+2+x^{2}$$

Add -70 and 2 to get -68.

$$5x-10=20x-68+x^{2}$$

Subtract 20x from both sides.

$$5x-10-20x=-68+x^{2}$$

Combine 5x and -20x to get -15x.

$$-15x-10=-68+x^{2}$$

Subtract -68 from both sides.

$$-15x-10-\left(-68\right)=x^{2}$$

The opposite of -68 is 68.

$$-15x-10+68=x^{2}$$

Subtract x^{2} from both sides.

$$-15x-10+68-x^{2}=0$$

Add -10 and 68 to get 58.

$$-15x+58-x^{2}=0$$

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.

$$-x^{2}-15x+58=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -15 for b, and 58 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-1\right)\times 58}}{2\left(-1\right)}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-1\right)\times 58}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-15\right)±\sqrt{225+4\times 58}}{2\left(-1\right)}$$

Multiply 4 times 58.

$$x=\frac{-\left(-15\right)±\sqrt{225+232}}{2\left(-1\right)}$$

Add 225 to 232.

$$x=\frac{-\left(-15\right)±\sqrt{457}}{2\left(-1\right)}$$

The opposite of -15 is 15.

$$x=\frac{15±\sqrt{457}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{15±\sqrt{457}}{-2}$$

Now solve the equation x=\frac{15±\sqrt{457}}{-2} when ± is plus. Add 15 to \sqrt{457}.

$$x=\frac{\sqrt{457}+15}{-2}$$

Divide 15+\sqrt{457} by -2.

$$x=\frac{-\sqrt{457}-15}{2}$$

Now solve the equation x=\frac{15±\sqrt{457}}{-2} when ± is minus. Subtract \sqrt{457} from 15.

$$x=\frac{15-\sqrt{457}}{-2}$$

Divide 15-\sqrt{457} by -2.

$$x=\frac{\sqrt{457}-15}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{457}-15}{2} x=\frac{\sqrt{457}-15}{2}$$

Steps for Completing the Square

Multiply 2 and 10 to get 20.

$$5x-10=20x-70+2+x^{2}$$

Add -70 and 2 to get -68.

$$5x-10=20x-68+x^{2}$$

Subtract 20x from both sides.

$$5x-10-20x=-68+x^{2}$$

Combine 5x and -20x to get -15x.

$$-15x-10=-68+x^{2}$$

Subtract x^{2} from both sides.

$$-15x-10-x^{2}=-68$$

Add 10 to both sides.

$$-15x-x^{2}=-68+10$$

Add -68 and 10 to get -58.

$$-15x-x^{2}=-58$$

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.

$$-x^{2}-15x=-58$$

Divide both sides by -1.

$$\frac{-x^{2}-15x}{-1}=-\frac{58}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{15}{-1}\right)x=-\frac{58}{-1}$$

Divide -15 by -1.

$$x^{2}+15x=-\frac{58}{-1}$$

Divide -58 by -1.

$$x^{2}+15x=58$$

Divide 15, the coefficient of the x term, by 2 to get \frac{15}{2}. Then add the square of \frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=58+\left(\frac{15}{2}\right)^{2}$$

Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+15x+\frac{225}{4}=58+\frac{225}{4}$$

Add 58 to \frac{225}{4}.

$$x^{2}+15x+\frac{225}{4}=\frac{457}{4}$$

Factor x^{2}+15x+\frac{225}{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}.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{457}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{457}{4}}$$

Simplify.

$$x+\frac{15}{2}=\frac{\sqrt{457}}{2} x+\frac{15}{2}=-\frac{\sqrt{457}}{2}$$

Subtract \frac{15}{2} from both sides of the equation.

$$x=\frac{\sqrt{457}-15}{2} x=\frac{-\sqrt{457}-15}{2}$$