Solve 21x^2=10-x

$$21 x ^ { 2 } = 10 - x$$

Solve for x

$$x=-\frac{5}{7}\approx -0.714285714$$

$$x=\frac{2}{3}\approx 0.666666667$$

Steps Using Factoring By Grouping

Subtract 10 from both sides.

$$21x^{2}-10=-x$$

Add x to both sides.

$$21x^{2}-10+x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$21x^{2}+x-10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 21x^{2}+ax+bx-10. To find a and b, set up a system to be solved.

$$a+b=1 ab=21\left(-10\right)=-210$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -210.

$$-1,210 -2,105 -3,70 -5,42 -6,35 -7,30 -10,21 -14,15$$

Calculate the sum for each pair.

$$-1+210=209 -2+105=103 -3+70=67 -5+42=37 -6+35=29 -7+30=23 -10+21=11 -14+15=1$$

The solution is the pair that gives sum 1.

$$a=-14 b=15$$

Rewrite 21x^{2}+x-10 as \left(21x^{2}-14x\right)+\left(15x-10\right).

$$\left(21x^{2}-14x\right)+\left(15x-10\right)$$

Factor out 7x in the first and 5 in the second group.

$$7x\left(3x-2\right)+5\left(3x-2\right)$$

Factor out common term 3x-2 by using distributive property.

$$\left(3x-2\right)\left(7x+5\right)$$

To find equation solutions, solve 3x-2=0 and 7x+5=0.

$$x=\frac{2}{3} x=-\frac{5}{7}$$

Steps Using the Quadratic Formula

Subtract 10 from both sides.

$$21x^{2}-10=-x$$

Add x to both sides.

$$21x^{2}-10+x=0$$

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.

$$21x^{2}+x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 21 for a, 1 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 21\left(-10\right)}}{2\times 21}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 21\left(-10\right)}}{2\times 21}$$

Multiply -4 times 21.

$$x=\frac{-1±\sqrt{1-84\left(-10\right)}}{2\times 21}$$

Multiply -84 times -10.

$$x=\frac{-1±\sqrt{1+840}}{2\times 21}$$

Add 1 to 840.

$$x=\frac{-1±\sqrt{841}}{2\times 21}$$

Take the square root of 841.

$$x=\frac{-1±29}{2\times 21}$$

Multiply 2 times 21.

$$x=\frac{-1±29}{42}$$

Now solve the equation x=\frac{-1±29}{42} when ± is plus. Add -1 to 29.

$$x=\frac{28}{42}$$

Reduce the fraction \frac{28}{42} to lowest terms by extracting and canceling out 14.

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

Now solve the equation x=\frac{-1±29}{42} when ± is minus. Subtract 29 from -1.

$$x=-\frac{30}{42}$$

Reduce the fraction \frac{-30}{42} to lowest terms by extracting and canceling out 6.

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

The equation is now solved.

$$x=\frac{2}{3} x=-\frac{5}{7}$$

Steps for Completing the Square

Add x to both sides.

$$21x^{2}+x=10$$

Divide both sides by 21.

$$\frac{21x^{2}+x}{21}=\frac{10}{21}$$

Dividing by 21 undoes the multiplication by 21.

$$x^{2}+\frac{1}{21}x=\frac{10}{21}$$

Divide \frac{1}{21}, the coefficient of the x term, by 2 to get \frac{1}{42}. Then add the square of \frac{1}{42} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{21}x+\left(\frac{1}{42}\right)^{2}=\frac{10}{21}+\left(\frac{1}{42}\right)^{2}$$

Square \frac{1}{42} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{21}x+\frac{1}{1764}=\frac{10}{21}+\frac{1}{1764}$$

Add \frac{10}{21} to \frac{1}{1764} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{21}x+\frac{1}{1764}=\frac{841}{1764}$$

Factor x^{2}+\frac{1}{21}x+\frac{1}{1764}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{42}\right)^{2}=\frac{841}{1764}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{42}\right)^{2}}=\sqrt{\frac{841}{1764}}$$

Simplify.

$$x+\frac{1}{42}=\frac{29}{42} x+\frac{1}{42}=-\frac{29}{42}$$

Subtract \frac{1}{42} from both sides of the equation.

$$x=\frac{2}{3} x=-\frac{5}{7}$$