Solve (x+7)*(3x+0,5)=7x*(x+3)

$$( x + 7 ) \cdot ( 3 x + 0,5 ) = 7 x \cdot ( x + 3 )$$

Solve for x

$$x=-0,875$$

$$x=1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+7 by 3x+0,5 and combine like terms.

$$3x^{2}+21,5x+3,5=7x\left(x+3\right)$$

Use the distributive property to multiply 7x by x+3.

$$3x^{2}+21,5x+3,5=7x^{2}+21x$$

Subtract 7x^{2} from both sides.

$$3x^{2}+21,5x+3,5-7x^{2}=21x$$

Combine 3x^{2} and -7x^{2} to get -4x^{2}.

$$-4x^{2}+21,5x+3,5=21x$$

Subtract 21x from both sides.

$$-4x^{2}+21,5x+3,5-21x=0$$

Combine 21,5x and -21x to get 0,5x.

$$-4x^{2}+0,5x+3,5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 0,5 for b, and 3,5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0,5±\sqrt{0,5^{2}-4\left(-4\right)\times 3,5}}{2\left(-4\right)}$$

Square 0,5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0,5±\sqrt{0,25-4\left(-4\right)\times 3,5}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-0,5±\sqrt{0,25+16\times 3,5}}{2\left(-4\right)}$$

Multiply 16 times 3,5.

$$x=\frac{-0,5±\sqrt{0,25+56}}{2\left(-4\right)}$$

Add 0,25 to 56.

$$x=\frac{-0,5±\sqrt{56,25}}{2\left(-4\right)}$$

Take the square root of 56,25.

$$x=\frac{-0,5±\frac{15}{2}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-0,5±\frac{15}{2}}{-8}$$

Now solve the equation x=\frac{-0,5±\frac{15}{2}}{-8} when ± is plus. Add -0,5 to \frac{15}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{7}{-8}$$

Divide 7 by -8.

$$x=-\frac{7}{8}$$

Now solve the equation x=\frac{-0,5±\frac{15}{2}}{-8} when ± is minus. Subtract \frac{15}{2} from -0,5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide -8 by -8.

$$x=1$$

The equation is now solved.

$$x=-\frac{7}{8} x=1$$

Steps for Completing the Square

Use the distributive property to multiply x+7 by 3x+0,5 and combine like terms.

$$3x^{2}+21,5x+3,5=7x\left(x+3\right)$$

Use the distributive property to multiply 7x by x+3.

$$3x^{2}+21,5x+3,5=7x^{2}+21x$$

Subtract 7x^{2} from both sides.

$$3x^{2}+21,5x+3,5-7x^{2}=21x$$

Combine 3x^{2} and -7x^{2} to get -4x^{2}.

$$-4x^{2}+21,5x+3,5=21x$$

Subtract 21x from both sides.

$$-4x^{2}+21,5x+3,5-21x=0$$

Combine 21,5x and -21x to get 0,5x.

$$-4x^{2}+0,5x+3,5=0$$

Subtract 3,5 from both sides. Anything subtracted from zero gives its negation.

$$-4x^{2}+0,5x=-3,5$$

Divide both sides by -4.

$$\frac{-4x^{2}+0,5x}{-4}=-\frac{3,5}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{0,5}{-4}x=-\frac{3,5}{-4}$$

Divide 0,5 by -4.

$$x^{2}-0,125x=-\frac{3,5}{-4}$$

Divide -3,5 by -4.

$$x^{2}-0,125x=0,875$$

Divide -0,125, the coefficient of the x term, by 2 to get -0,0625. Then add the square of -0,0625 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-0,125x+\left(-0,0625\right)^{2}=0,875+\left(-0,0625\right)^{2}$$

Square -0,0625 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-0,125x+0,00390625=0,875+0,00390625$$

Add 0,875 to 0,00390625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-0,125x+0,00390625=0,87890625$$

Factor x^{2}-0,125x+0,00390625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0,0625\right)^{2}=0,87890625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0,0625\right)^{2}}=\sqrt{0,87890625}$$

Simplify.

$$x-0,0625=\frac{15}{16} x-0,0625=-\frac{15}{16}$$

Add 0,0625 to both sides of the equation.

$$x=1 x=-\frac{7}{8}$$