Solve 10x^2=2.5left(x-5right)^2

$$10 { x }^{ 2 } =2.5 { \left(x-5 \right) }^{ 2 }$$

Solve for x

$$x=-5$$

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$10x^{2}=2.5\left(x^{2}-10x+25\right)$$

Use the distributive property to multiply 2.5 by x^{2}-10x+25.

$$10x^{2}=2.5x^{2}-25x+62.5$$

Subtract 2.5x^{2} from both sides.

$$10x^{2}-2.5x^{2}=-25x+62.5$$

Combine 10x^{2} and -2.5x^{2} to get 7.5x^{2}.

$$7.5x^{2}=-25x+62.5$$

Add 25x to both sides.

$$7.5x^{2}+25x=62.5$$

Subtract 62.5 from both sides.

$$7.5x^{2}+25x-62.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7.5 for a, 25 for b, and -62.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\times 7.5\left(-62.5\right)}}{2\times 7.5}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\times 7.5\left(-62.5\right)}}{2\times 7.5}$$

Multiply -4 times 7.5.

$$x=\frac{-25±\sqrt{625-30\left(-62.5\right)}}{2\times 7.5}$$

Multiply -30 times -62.5.

$$x=\frac{-25±\sqrt{625+1875}}{2\times 7.5}$$

Add 625 to 1875.

$$x=\frac{-25±\sqrt{2500}}{2\times 7.5}$$

Take the square root of 2500.

$$x=\frac{-25±50}{2\times 7.5}$$

Multiply 2 times 7.5.

$$x=\frac{-25±50}{15}$$

Now solve the equation x=\frac{-25±50}{15} when ± is plus. Add -25 to 50.

$$x=\frac{25}{15}$$

Reduce the fraction \frac{25}{15} to lowest terms by extracting and canceling out 5.

$$x=\frac{5}{3}$$

Now solve the equation x=\frac{-25±50}{15} when ± is minus. Subtract 50 from -25.

$$x=-\frac{75}{15}$$

Divide -75 by 15.

$$x=-5$$

The equation is now solved.

$$x=\frac{5}{3} x=-5$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$10x^{2}=2.5\left(x^{2}-10x+25\right)$$

Use the distributive property to multiply 2.5 by x^{2}-10x+25.

$$10x^{2}=2.5x^{2}-25x+62.5$$

Subtract 2.5x^{2} from both sides.

$$10x^{2}-2.5x^{2}=-25x+62.5$$

Combine 10x^{2} and -2.5x^{2} to get 7.5x^{2}.

$$7.5x^{2}=-25x+62.5$$

Add 25x to both sides.

$$7.5x^{2}+25x=62.5$$

Divide both sides of the equation by 7.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{7.5x^{2}+25x}{7.5}=\frac{62.5}{7.5}$$

Dividing by 7.5 undoes the multiplication by 7.5.

$$x^{2}+\frac{25}{7.5}x=\frac{62.5}{7.5}$$

Divide 25 by 7.5 by multiplying 25 by the reciprocal of 7.5.

$$x^{2}+\frac{10}{3}x=\frac{62.5}{7.5}$$

Divide 62.5 by 7.5 by multiplying 62.5 by the reciprocal of 7.5.

$$x^{2}+\frac{10}{3}x=\frac{25}{3}$$

Divide \frac{10}{3}, the coefficient of the x term, by 2 to get \frac{5}{3}. Then add the square of \frac{5}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{10}{3}x+\frac{5}{3}^{2}=\frac{25}{3}+\frac{5}{3}^{2}$$

Square \frac{5}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{10}{3}x+\frac{25}{9}=\frac{25}{3}+\frac{25}{9}$$

Add \frac{25}{3} to \frac{25}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{10}{3}x+\frac{25}{9}=\frac{100}{9}$$

Factor x^{2}+\frac{10}{3}x+\frac{25}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{3}\right)^{2}=\frac{100}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{3}\right)^{2}}=\sqrt{\frac{100}{9}}$$

Simplify.

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

Subtract \frac{5}{3} from both sides of the equation.

$$x=\frac{5}{3} x=-5$$