Solve 2.4^2=2.5^2-left(1.5-xright)^2

$${ 2.4 }^{ 2 } = { 2.5 }^{ 2 } - { \left(1.5-x \right) }^{ 2 }$$

Solve for x

$$x=0.8$$

$$x=2.2$$

Steps Using the Quadratic Formula

Calculate 2.4 to the power of 2 and get 5.76.

$$5.76=2.5^{2}-\left(1.5-x\right)^{2}$$

Calculate 2.5 to the power of 2 and get 6.25.

$$5.76=6.25-\left(1.5-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1.5-x\right)^{2}.

$$5.76=6.25-\left(2.25-3x+x^{2}\right)$$

To find the opposite of 2.25-3x+x^{2}, find the opposite of each term.

$$5.76=6.25-2.25+3x-x^{2}$$

Subtract 2.25 from 6.25 to get 4.

$$5.76=4+3x-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$4+3x-x^{2}=5.76$$

Subtract 5.76 from both sides.

$$4+3x-x^{2}-5.76=0$$

Subtract 5.76 from 4 to get -1.76.

$$-1.76+3x-x^{2}=0$$

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.

$$-x^{2}+3x-1.76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 3 for b, and -1.76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\left(-1.76\right)}}{2\left(-1\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-1\right)\left(-1.76\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-3±\sqrt{9+4\left(-1.76\right)}}{2\left(-1\right)}$$

Multiply 4 times -1.76.

$$x=\frac{-3±\sqrt{9-7.04}}{2\left(-1\right)}$$

Add 9 to -7.04.

$$x=\frac{-3±\sqrt{1.96}}{2\left(-1\right)}$$

Take the square root of 1.96.

$$x=\frac{-3±\frac{7}{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-3±\frac{7}{5}}{-2}$$

Now solve the equation x=\frac{-3±\frac{7}{5}}{-2} when ± is plus. Add -3 to \frac{7}{5}.

$$x=-\frac{\frac{8}{5}}{-2}$$

Divide -\frac{8}{5} by -2.

$$x=\frac{4}{5}$$

Now solve the equation x=\frac{-3±\frac{7}{5}}{-2} when ± is minus. Subtract \frac{7}{5} from -3.

$$x=-\frac{\frac{22}{5}}{-2}$$

Divide -\frac{22}{5} by -2.

$$x=\frac{11}{5}$$

The equation is now solved.

$$x=\frac{4}{5} x=\frac{11}{5}$$

Steps for Completing the Square

Calculate 2.4 to the power of 2 and get 5.76.

$$5.76=2.5^{2}-\left(1.5-x\right)^{2}$$

Calculate 2.5 to the power of 2 and get 6.25.

$$5.76=6.25-\left(1.5-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1.5-x\right)^{2}.

$$5.76=6.25-\left(2.25-3x+x^{2}\right)$$

To find the opposite of 2.25-3x+x^{2}, find the opposite of each term.

$$5.76=6.25-2.25+3x-x^{2}$$

Subtract 2.25 from 6.25 to get 4.

$$5.76=4+3x-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$4+3x-x^{2}=5.76$$

Subtract 4 from both sides.

$$3x-x^{2}=5.76-4$$

Subtract 4 from 5.76 to get 1.76.

$$3x-x^{2}=1.76$$

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.

$$-x^{2}+3x=1.76$$

Divide both sides by -1.

$$\frac{-x^{2}+3x}{-1}=\frac{1.76}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{3}{-1}x=\frac{1.76}{-1}$$

Divide 3 by -1.

$$x^{2}-3x=\frac{1.76}{-1}$$

Divide 1.76 by -1.

$$x^{2}-3x=-1.76$$

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

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=-1.76+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=-1.76+\frac{9}{4}$$

Add -1.76 to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-3x+\frac{9}{4}=\frac{49}{100}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{49}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{100}}$$

Simplify.

$$x-\frac{3}{2}=\frac{7}{10} x-\frac{3}{2}=-\frac{7}{10}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{11}{5} x=\frac{4}{5}$$