Solve 25x^2-10x-x^2-25=0

$$25 x ^ { 2 } - 10 x - x ^ { 2 } - 25 = 0$$

Solve for x

$$x=-\frac{5}{6}\approx -0.833333333$$

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Steps Using Factoring By Grouping

Combine 25x^{2} and -x^{2} to get 24x^{2}.

$$24x^{2}-10x-25=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 24x^{2}+ax+bx-25. To find a and b, set up a system to be solved.

$$a+b=-10 ab=24\left(-25\right)=-600$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -600.

$$1,-600 2,-300 3,-200 4,-150 5,-120 6,-100 8,-75 10,-60 12,-50 15,-40 20,-30 24,-25$$

Calculate the sum for each pair.

$$1-600=-599 2-300=-298 3-200=-197 4-150=-146 5-120=-115 6-100=-94 8-75=-67 10-60=-50 12-50=-38 15-40=-25 20-30=-10 24-25=-1$$

The solution is the pair that gives sum -10.

$$a=-30 b=20$$

Rewrite 24x^{2}-10x-25 as \left(24x^{2}-30x\right)+\left(20x-25\right).

$$\left(24x^{2}-30x\right)+\left(20x-25\right)$$

Factor out 6x in the first and 5 in the second group.

$$6x\left(4x-5\right)+5\left(4x-5\right)$$

Factor out common term 4x-5 by using distributive property.

$$\left(4x-5\right)\left(6x+5\right)$$

To find equation solutions, solve 4x-5=0 and 6x+5=0.

$$x=\frac{5}{4} x=-\frac{5}{6}$$

Steps Using the Quadratic Formula

Combine 25x^{2} and -x^{2} to get 24x^{2}.

$$24x^{2}-10x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, -10 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 24\left(-25\right)}}{2\times 24}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 24\left(-25\right)}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-\left(-10\right)±\sqrt{100-96\left(-25\right)}}{2\times 24}$$

Multiply -96 times -25.

$$x=\frac{-\left(-10\right)±\sqrt{100+2400}}{2\times 24}$$

Add 100 to 2400.

$$x=\frac{-\left(-10\right)±\sqrt{2500}}{2\times 24}$$

Take the square root of 2500.

$$x=\frac{-\left(-10\right)±50}{2\times 24}$$

The opposite of -10 is 10.

$$x=\frac{10±50}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{10±50}{48}$$

Now solve the equation x=\frac{10±50}{48} when ± is plus. Add 10 to 50.

$$x=\frac{60}{48}$$

Reduce the fraction \frac{60}{48} to lowest terms by extracting and canceling out 12.

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

Now solve the equation x=\frac{10±50}{48} when ± is minus. Subtract 50 from 10.

$$x=-\frac{40}{48}$$

Reduce the fraction \frac{-40}{48} to lowest terms by extracting and canceling out 8.

$$x=-\frac{5}{6}$$

The equation is now solved.

$$x=\frac{5}{4} x=-\frac{5}{6}$$

Steps for Completing the Square

Combine 25x^{2} and -x^{2} to get 24x^{2}.

$$24x^{2}-10x-25=0$$

Add 25 to both sides. Anything plus zero gives itself.

$$24x^{2}-10x=25$$

Divide both sides by 24.

$$\frac{24x^{2}-10x}{24}=\frac{25}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$x^{2}+\left(-\frac{10}{24}\right)x=\frac{25}{24}$$

Reduce the fraction \frac{-10}{24} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{5}{12}x=\frac{25}{24}$$

Divide -\frac{5}{12}, the coefficient of the x term, by 2 to get -\frac{5}{24}. Then add the square of -\frac{5}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{12}x+\left(-\frac{5}{24}\right)^{2}=\frac{25}{24}+\left(-\frac{5}{24}\right)^{2}$$

Square -\frac{5}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{12}x+\frac{25}{576}=\frac{25}{24}+\frac{25}{576}$$

Add \frac{25}{24} to \frac{25}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{12}x+\frac{25}{576}=\frac{625}{576}$$

Factor x^{2}-\frac{5}{12}x+\frac{25}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{24}\right)^{2}=\frac{625}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{24}\right)^{2}}=\sqrt{\frac{625}{576}}$$

Simplify.

$$x-\frac{5}{24}=\frac{25}{24} x-\frac{5}{24}=-\frac{25}{24}$$

Add \frac{5}{24} to both sides of the equation.

$$x=\frac{5}{4} x=-\frac{5}{6}$$