Solve (48-x)(52-x)=2300

$$(48-x)(52-x)=2300$$

Solve for x

$$x=98$$

$$x=2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 48-x by 52-x and combine like terms.

$$2496-100x+x^{2}=2300$$

Subtract 2300 from both sides.

$$2496-100x+x^{2}-2300=0$$

Subtract 2300 from 2496 to get 196.

$$196-100x+x^{2}=0$$

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.

$$x^{2}-100x+196=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -100 for b, and 196 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-100\right)±\sqrt{\left(-100\right)^{2}-4\times 196}}{2}$$

Square -100.

$$x=\frac{-\left(-100\right)±\sqrt{10000-4\times 196}}{2}$$

Multiply -4 times 196.

$$x=\frac{-\left(-100\right)±\sqrt{10000-784}}{2}$$

Add 10000 to -784.

$$x=\frac{-\left(-100\right)±\sqrt{9216}}{2}$$

Take the square root of 9216.

$$x=\frac{-\left(-100\right)±96}{2}$$

The opposite of -100 is 100.

$$x=\frac{100±96}{2}$$

Now solve the equation x=\frac{100±96}{2} when ± is plus. Add 100 to 96.

$$x=\frac{196}{2}$$

Divide 196 by 2.

$$x=98$$

Now solve the equation x=\frac{100±96}{2} when ± is minus. Subtract 96 from 100.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

The equation is now solved.

$$x=98 x=2$$

Steps for Completing the Square

Use the distributive property to multiply 48-x by 52-x and combine like terms.

$$2496-100x+x^{2}=2300$$

Subtract 2496 from both sides.

$$-100x+x^{2}=2300-2496$$

Subtract 2496 from 2300 to get -196.

$$-100x+x^{2}=-196$$

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.

$$x^{2}-100x=-196$$

Divide -100, the coefficient of the x term, by 2 to get -50. Then add the square of -50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-100x+\left(-50\right)^{2}=-196+\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=-196+2500$$

Add -196 to 2500.

$$x^{2}-100x+2500=2304$$

Factor x^{2}-100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50\right)^{2}=2304$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{2304}$$

Simplify.

$$x-50=48 x-50=-48$$

Add 50 to both sides of the equation.

$$x=98 x=2$$