Solve 7a(5-2a)=14

$$7 a ( 5 - 2 a ) = 14$$

Solve for a

$$a=2$$

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

Steps Using the Quadratic Formula

Use the distributive property to multiply 7a by 5-2a.

$$35a-14a^{2}=14$$

Subtract 14 from both sides.

$$35a-14a^{2}-14=0$$

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.

$$-14a^{2}+35a-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -14 for a, 35 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-35±\sqrt{35^{2}-4\left(-14\right)\left(-14\right)}}{2\left(-14\right)}$$

Square 35.

$$a=\frac{-35±\sqrt{1225-4\left(-14\right)\left(-14\right)}}{2\left(-14\right)}$$

Multiply -4 times -14.

$$a=\frac{-35±\sqrt{1225+56\left(-14\right)}}{2\left(-14\right)}$$

Multiply 56 times -14.

$$a=\frac{-35±\sqrt{1225-784}}{2\left(-14\right)}$$

Add 1225 to -784.

$$a=\frac{-35±\sqrt{441}}{2\left(-14\right)}$$

Take the square root of 441.

$$a=\frac{-35±21}{2\left(-14\right)}$$

Multiply 2 times -14.

$$a=\frac{-35±21}{-28}$$

Now solve the equation a=\frac{-35±21}{-28} when ± is plus. Add -35 to 21.

$$a=-\frac{14}{-28}$$

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

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

Now solve the equation a=\frac{-35±21}{-28} when ± is minus. Subtract 21 from -35.

$$a=-\frac{56}{-28}$$

Divide -56 by -28.

$$a=2$$

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply 7a by 5-2a.

$$35a-14a^{2}=14$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-14a^{2}+35a=14$$

Divide both sides by -14.

$$\frac{-14a^{2}+35a}{-14}=\frac{14}{-14}$$

Dividing by -14 undoes the multiplication by -14.

$$a^{2}+\frac{35}{-14}a=\frac{14}{-14}$$

Reduce the fraction \frac{35}{-14} to lowest terms by extracting and canceling out 7.

$$a^{2}-\frac{5}{2}a=\frac{14}{-14}$$

Divide 14 by -14.

$$a^{2}-\frac{5}{2}a=-1$$

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.

$$a^{2}-\frac{5}{2}a+\left(-\frac{5}{4}\right)^{2}=-1+\left(-\frac{5}{4}\right)^{2}$$

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

$$a^{2}-\frac{5}{2}a+\frac{25}{16}=-1+\frac{25}{16}$$

Add -1 to \frac{25}{16}.

$$a^{2}-\frac{5}{2}a+\frac{25}{16}=\frac{9}{16}$$

Factor a^{2}-\frac{5}{2}a+\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}.

$$\left(a-\frac{5}{4}\right)^{2}=\frac{9}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{5}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Simplify.

$$a-\frac{5}{4}=\frac{3}{4} a-\frac{5}{4}=-\frac{3}{4}$$

Add \frac{5}{4} to both sides of the equation.

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