Solve x^2-45x/4+25/2=0

$$x ^ { 2 } - \frac { 45 x } { 4 } + \frac { 25 } { 2 } = 0$$

Solve for x

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

$$x=10$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$4x^{2}-45x+50=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx+50. To find a and b, set up a system to be solved.

$$a+b=-45 ab=4\times 50=200$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 200.

$$-1,-200 -2,-100 -4,-50 -5,-40 -8,-25 -10,-20$$

Calculate the sum for each pair.

$$-1-200=-201 -2-100=-102 -4-50=-54 -5-40=-45 -8-25=-33 -10-20=-30$$

The solution is the pair that gives sum -45.

$$a=-40 b=-5$$

Rewrite 4x^{2}-45x+50 as \left(4x^{2}-40x\right)+\left(-5x+50\right).

$$\left(4x^{2}-40x\right)+\left(-5x+50\right)$$

Factor out 4x in the first and -5 in the second group.

$$4x\left(x-10\right)-5\left(x-10\right)$$

Factor out common term x-10 by using distributive property.

$$\left(x-10\right)\left(4x-5\right)$$

To find equation solutions, solve x-10=0 and 4x-5=0.

$$x=10 x=\frac{5}{4}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$4x^{2}-45x+50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -45 for b, and 50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-45\right)±\sqrt{\left(-45\right)^{2}-4\times 4\times 50}}{2\times 4}$$

Square -45.

$$x=\frac{-\left(-45\right)±\sqrt{2025-4\times 4\times 50}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-45\right)±\sqrt{2025-16\times 50}}{2\times 4}$$

Multiply -16 times 50.

$$x=\frac{-\left(-45\right)±\sqrt{2025-800}}{2\times 4}$$

Add 2025 to -800.

$$x=\frac{-\left(-45\right)±\sqrt{1225}}{2\times 4}$$

Take the square root of 1225.

$$x=\frac{-\left(-45\right)±35}{2\times 4}$$

The opposite of -45 is 45.

$$x=\frac{45±35}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{45±35}{8}$$

Now solve the equation x=\frac{45±35}{8} when ± is plus. Add 45 to 35.

$$x=\frac{80}{8}$$

Divide 80 by 8.

$$x=10$$

Now solve the equation x=\frac{45±35}{8} when ± is minus. Subtract 35 from 45.

$$x=\frac{10}{8}$$

Reduce the fraction \frac{10}{8} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

$$x=10 x=\frac{5}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$4x^{2}-45x+50=0$$

Subtract 50 from both sides. Anything subtracted from zero gives its negation.

$$4x^{2}-45x=-50$$

Divide both sides by 4.

$$\frac{4x^{2}-45x}{4}=-\frac{50}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{45}{4}x=-\frac{50}{4}$$

Reduce the fraction \frac{-50}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{45}{4}x=-\frac{25}{2}$$

Divide -\frac{45}{4}, the coefficient of the x term, by 2 to get -\frac{45}{8}. Then add the square of -\frac{45}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{45}{4}x+\left(-\frac{45}{8}\right)^{2}=-\frac{25}{2}+\left(-\frac{45}{8}\right)^{2}$$

Square -\frac{45}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{45}{4}x+\frac{2025}{64}=-\frac{25}{2}+\frac{2025}{64}$$

Add -\frac{25}{2} to \frac{2025}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{45}{4}x+\frac{2025}{64}=\frac{1225}{64}$$

Factor x^{2}-\frac{45}{4}x+\frac{2025}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{45}{8}\right)^{2}=\frac{1225}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{45}{8}\right)^{2}}=\sqrt{\frac{1225}{64}}$$

Simplify.

$$x-\frac{45}{8}=\frac{35}{8} x-\frac{45}{8}=-\frac{35}{8}$$

Add \frac{45}{8} to both sides of the equation.

$$x=10 x=\frac{5}{4}$$