Solve 5xleft(2-xright)=-3left(x-1right)

$$5x \left( 2-x \right) = -3 \left( x-1 \right)$$

Solve for x

$$x = \frac{\sqrt{109} + 13}{10} \approx 2.344030651$$

$$x=\frac{13-\sqrt{109}}{10}\approx 0.255969349$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5x by 2-x.

$$10x-5x^{2}=-3\left(x-1\right)$$

Use the distributive property to multiply -3 by x-1.

$$10x-5x^{2}=-3x+3$$

Add 3x to both sides.

$$10x-5x^{2}+3x=3$$

Combine 10x and 3x to get 13x.

$$13x-5x^{2}=3$$

Subtract 3 from both sides.

$$13x-5x^{2}-3=0$$

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.

$$-5x^{2}+13x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 13 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-5\right)\left(-3\right)}}{2\left(-5\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-5\right)\left(-3\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-13±\sqrt{169+20\left(-3\right)}}{2\left(-5\right)}$$

Multiply 20 times -3.

$$x=\frac{-13±\sqrt{169-60}}{2\left(-5\right)}$$

Add 169 to -60.

$$x=\frac{-13±\sqrt{109}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-13±\sqrt{109}}{-10}$$

Now solve the equation x=\frac{-13±\sqrt{109}}{-10} when ± is plus. Add -13 to \sqrt{109}.

$$x=\frac{\sqrt{109}-13}{-10}$$

Divide -13+\sqrt{109} by -10.

$$x=\frac{13-\sqrt{109}}{10}$$

Now solve the equation x=\frac{-13±\sqrt{109}}{-10} when ± is minus. Subtract \sqrt{109} from -13.

$$x=\frac{-\sqrt{109}-13}{-10}$$

Divide -13-\sqrt{109} by -10.

$$x=\frac{\sqrt{109}+13}{10}$$

The equation is now solved.

$$x=\frac{13-\sqrt{109}}{10} x=\frac{\sqrt{109}+13}{10}$$

Steps for Completing the Square

Use the distributive property to multiply 5x by 2-x.

$$10x-5x^{2}=-3\left(x-1\right)$$

Use the distributive property to multiply -3 by x-1.

$$10x-5x^{2}=-3x+3$$

Add 3x to both sides.

$$10x-5x^{2}+3x=3$$

Combine 10x and 3x to get 13x.

$$13x-5x^{2}=3$$

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.

$$-5x^{2}+13x=3$$

Divide both sides by -5.

$$\frac{-5x^{2}+13x}{-5}=\frac{3}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{13}{-5}x=\frac{3}{-5}$$

Divide 13 by -5.

$$x^{2}-\frac{13}{5}x=\frac{3}{-5}$$

Divide 3 by -5.

$$x^{2}-\frac{13}{5}x=-\frac{3}{5}$$

Divide -\frac{13}{5}, the coefficient of the x term, by 2 to get -\frac{13}{10}. Then add the square of -\frac{13}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{5}x+\left(-\frac{13}{10}\right)^{2}=-\frac{3}{5}+\left(-\frac{13}{10}\right)^{2}$$

Square -\frac{13}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{5}x+\frac{169}{100}=-\frac{3}{5}+\frac{169}{100}$$

Add -\frac{3}{5} to \frac{169}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{5}x+\frac{169}{100}=\frac{109}{100}$$

Factor x^{2}-\frac{13}{5}x+\frac{169}{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}.

$$\left(x-\frac{13}{10}\right)^{2}=\frac{109}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{10}\right)^{2}}=\sqrt{\frac{109}{100}}$$

Simplify.

$$x-\frac{13}{10}=\frac{\sqrt{109}}{10} x-\frac{13}{10}=-\frac{\sqrt{109}}{10}$$

Add \frac{13}{10} to both sides of the equation.

$$x=\frac{\sqrt{109}+13}{10} x=\frac{13-\sqrt{109}}{10}$$