Solve -4x^2=10x-6

$$- 4 x ^ { 2 } = 10 x - 6$$

Solve for x

$$x=\frac{1}{2}=0.5$$

$$x=-3$$

Steps Using Factoring By Grouping

Subtract 10x from both sides.

$$-4x^{2}-10x=-6$$

Add 6 to both sides.

$$-4x^{2}-10x+6=0$$

Divide both sides by 2.

$$-2x^{2}-5x+3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

$$a+b=-5 ab=-2\times 3=-6$$

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 -6.

$$1,-6 2,-3$$

Calculate the sum for each pair.

$$1-6=-5 2-3=-1$$

The solution is the pair that gives sum -5.

$$a=1 b=-6$$

Rewrite -2x^{2}-5x+3 as \left(-2x^{2}+x\right)+\left(-6x+3\right).

$$\left(-2x^{2}+x\right)+\left(-6x+3\right)$$

Factor out -x in the first and -3 in the second group.

$$-x\left(2x-1\right)-3\left(2x-1\right)$$

Factor out common term 2x-1 by using distributive property.

$$\left(2x-1\right)\left(-x-3\right)$$

To find equation solutions, solve 2x-1=0 and -x-3=0.

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

Steps Using the Quadratic Formula

Subtract 10x from both sides.

$$-4x^{2}-10x=-6$$

Add 6 to both sides.

$$-4x^{2}-10x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -10 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-4\right)\times 6}}{2\left(-4\right)}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-4\right)\times 6}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-10\right)±\sqrt{100+16\times 6}}{2\left(-4\right)}$$

Multiply 16 times 6.

$$x=\frac{-\left(-10\right)±\sqrt{100+96}}{2\left(-4\right)}$$

Add 100 to 96.

$$x=\frac{-\left(-10\right)±\sqrt{196}}{2\left(-4\right)}$$

Take the square root of 196.

$$x=\frac{-\left(-10\right)±14}{2\left(-4\right)}$$

The opposite of -10 is 10.

$$x=\frac{10±14}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{10±14}{-8}$$

Now solve the equation x=\frac{10±14}{-8} when ± is plus. Add 10 to 14.

$$x=\frac{24}{-8}$$

Divide 24 by -8.

$$x=-3$$

Now solve the equation x=\frac{10±14}{-8} when ± is minus. Subtract 14 from 10.

$$x=-\frac{4}{-8}$$

Reduce the fraction \frac{-4}{-8} to lowest terms by extracting and canceling out 4.

$$x=\frac{1}{2}$$

The equation is now solved.

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

Steps for Completing the Square

Subtract 10x from both sides.

$$-4x^{2}-10x=-6$$

Divide both sides by -4.

$$\frac{-4x^{2}-10x}{-4}=-\frac{6}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\left(-\frac{10}{-4}\right)x=-\frac{6}{-4}$$

Reduce the fraction \frac{-10}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{5}{2}x=-\frac{6}{-4}$$

Reduce the fraction \frac{-6}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{5}{2}x=\frac{3}{2}$$

Divide \frac{5}{2}, the coefficient of the x term, by 2 to get \frac{5}{4}. Then add the square of \frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{2}x+\left(\frac{5}{4}\right)^{2}=\frac{3}{2}+\left(\frac{5}{4}\right)^{2}$$

Square \frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{3}{2}+\frac{25}{16}$$

Add \frac{3}{2} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{49}{16}$$

Factor x^{2}+\frac{5}{2}x+\frac{25}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x+\frac{5}{4}=\frac{7}{4} x+\frac{5}{4}=-\frac{7}{4}$$

Subtract \frac{5}{4} from both sides of the equation.

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