Solve 4(x-3)^2-25(x-2)^2=0

$$4 ( x - 3 ) ^ { 2 } - 25 ( x - 2 ) ^ { 2 } = 0$$

Solve for x

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

$$x = \frac{16}{7} = 2\frac{2}{7} \approx 2.285714286$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$4\left(x^{2}-6x+9\right)-25\left(x-2\right)^{2}=0$$

Use the distributive property to multiply 4 by x^{2}-6x+9.

$$4x^{2}-24x+36-25\left(x-2\right)^{2}=0$$

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

$$4x^{2}-24x+36-25\left(x^{2}-4x+4\right)=0$$

Use the distributive property to multiply -25 by x^{2}-4x+4.

$$4x^{2}-24x+36-25x^{2}+100x-100=0$$

Combine 4x^{2} and -25x^{2} to get -21x^{2}.

$$-21x^{2}-24x+36+100x-100=0$$

Combine -24x and 100x to get 76x.

$$-21x^{2}+76x+36-100=0$$

Subtract 100 from 36 to get -64.

$$-21x^{2}+76x-64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -21 for a, 76 for b, and -64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-76±\sqrt{76^{2}-4\left(-21\right)\left(-64\right)}}{2\left(-21\right)}$$

Square 76.

$$x=\frac{-76±\sqrt{5776-4\left(-21\right)\left(-64\right)}}{2\left(-21\right)}$$

Multiply -4 times -21.

$$x=\frac{-76±\sqrt{5776+84\left(-64\right)}}{2\left(-21\right)}$$

Multiply 84 times -64.

$$x=\frac{-76±\sqrt{5776-5376}}{2\left(-21\right)}$$

Add 5776 to -5376.

$$x=\frac{-76±\sqrt{400}}{2\left(-21\right)}$$

Take the square root of 400.

$$x=\frac{-76±20}{2\left(-21\right)}$$

Multiply 2 times -21.

$$x=\frac{-76±20}{-42}$$

Now solve the equation x=\frac{-76±20}{-42} when ± is plus. Add -76 to 20.

$$x=-\frac{56}{-42}$$

Reduce the fraction \frac{-56}{-42} to lowest terms by extracting and canceling out 14.

$$x=\frac{4}{3}$$

Now solve the equation x=\frac{-76±20}{-42} when ± is minus. Subtract 20 from -76.

$$x=-\frac{96}{-42}$$

Reduce the fraction \frac{-96}{-42} to lowest terms by extracting and canceling out 6.

$$x=\frac{16}{7}$$

The equation is now solved.

$$x=\frac{4}{3} x=\frac{16}{7}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$4\left(x^{2}-6x+9\right)-25\left(x-2\right)^{2}=0$$

Use the distributive property to multiply 4 by x^{2}-6x+9.

$$4x^{2}-24x+36-25\left(x-2\right)^{2}=0$$

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

$$4x^{2}-24x+36-25\left(x^{2}-4x+4\right)=0$$

Use the distributive property to multiply -25 by x^{2}-4x+4.

$$4x^{2}-24x+36-25x^{2}+100x-100=0$$

Combine 4x^{2} and -25x^{2} to get -21x^{2}.

$$-21x^{2}-24x+36+100x-100=0$$

Combine -24x and 100x to get 76x.

$$-21x^{2}+76x+36-100=0$$

Subtract 100 from 36 to get -64.

$$-21x^{2}+76x-64=0$$

Add 64 to both sides. Anything plus zero gives itself.

$$-21x^{2}+76x=64$$

Divide both sides by -21.

$$\frac{-21x^{2}+76x}{-21}=\frac{64}{-21}$$

Dividing by -21 undoes the multiplication by -21.

$$x^{2}+\frac{76}{-21}x=\frac{64}{-21}$$

Divide 76 by -21.

$$x^{2}-\frac{76}{21}x=\frac{64}{-21}$$

Divide 64 by -21.

$$x^{2}-\frac{76}{21}x=-\frac{64}{21}$$

Divide -\frac{76}{21}, the coefficient of the x term, by 2 to get -\frac{38}{21}. Then add the square of -\frac{38}{21} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{76}{21}x+\left(-\frac{38}{21}\right)^{2}=-\frac{64}{21}+\left(-\frac{38}{21}\right)^{2}$$

Square -\frac{38}{21} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{76}{21}x+\frac{1444}{441}=-\frac{64}{21}+\frac{1444}{441}$$

Add -\frac{64}{21} to \frac{1444}{441} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{76}{21}x+\frac{1444}{441}=\frac{100}{441}$$

Factor x^{2}-\frac{76}{21}x+\frac{1444}{441}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{38}{21}\right)^{2}=\frac{100}{441}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{38}{21}\right)^{2}}=\sqrt{\frac{100}{441}}$$

Simplify.

$$x-\frac{38}{21}=\frac{10}{21} x-\frac{38}{21}=-\frac{10}{21}$$

Add \frac{38}{21} to both sides of the equation.

$$x=\frac{16}{7} x=\frac{4}{3}$$