Solve 8-6y/7*-7-4y/3=2

$$\frac { 8 - 6 y } { 7 } \cdot \frac { - 7 - 4 y } { 3 } = 2$$

Solve for y

$$y = \frac{\sqrt{2377} - 5}{24} \approx 1.823103624$$

$$y=\frac{-\sqrt{2377}-5}{24}\approx -2.239770291$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 21, the least common multiple of 7,3.

$$\left(8-6y\right)\left(-7-4y\right)=42$$

Apply the distributive property by multiplying each term of 8-6y by each term of -7-4y.

$$-56-32y+42y+24y^{2}=42$$

Combine -32y and 42y to get 10y.

$$-56+10y+24y^{2}=42$$

Subtract 42 from both sides.

$$-56+10y+24y^{2}-42=0$$

Subtract 42 from -56 to get -98.

$$-98+10y+24y^{2}=0$$

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.

$$24y^{2}+10y-98=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, 10 for b, and -98 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-10±\sqrt{10^{2}-4\times 24\left(-98\right)}}{2\times 24}$$

Square 10.

$$y=\frac{-10±\sqrt{100-4\times 24\left(-98\right)}}{2\times 24}$$

Multiply -4 times 24.

$$y=\frac{-10±\sqrt{100-96\left(-98\right)}}{2\times 24}$$

Multiply -96 times -98.

$$y=\frac{-10±\sqrt{100+9408}}{2\times 24}$$

Add 100 to 9408.

$$y=\frac{-10±\sqrt{9508}}{2\times 24}$$

Take the square root of 9508.

$$y=\frac{-10±2\sqrt{2377}}{2\times 24}$$

Multiply 2 times 24.

$$y=\frac{-10±2\sqrt{2377}}{48}$$

Now solve the equation y=\frac{-10±2\sqrt{2377}}{48} when ± is plus. Add -10 to 2\sqrt{2377}.

$$y=\frac{2\sqrt{2377}-10}{48}$$

Divide -10+2\sqrt{2377} by 48.

$$y=\frac{\sqrt{2377}-5}{24}$$

Now solve the equation y=\frac{-10±2\sqrt{2377}}{48} when ± is minus. Subtract 2\sqrt{2377} from -10.

$$y=\frac{-2\sqrt{2377}-10}{48}$$

Divide -10-2\sqrt{2377} by 48.

$$y=\frac{-\sqrt{2377}-5}{24}$$

The equation is now solved.

$$y=\frac{\sqrt{2377}-5}{24} y=\frac{-\sqrt{2377}-5}{24}$$

Steps for Completing the Square

Multiply both sides of the equation by 21, the least common multiple of 7,3.

$$\left(8-6y\right)\left(-7-4y\right)=42$$

Apply the distributive property by multiplying each term of 8-6y by each term of -7-4y.

$$-56-32y+42y+24y^{2}=42$$

Combine -32y and 42y to get 10y.

$$-56+10y+24y^{2}=42$$

Add 56 to both sides.

$$10y+24y^{2}=42+56$$

Add 42 and 56 to get 98.

$$10y+24y^{2}=98$$

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.

$$24y^{2}+10y=98$$

Divide both sides by 24.

$$\frac{24y^{2}+10y}{24}=\frac{98}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$y^{2}+\frac{10}{24}y=\frac{98}{24}$$

Reduce the fraction \frac{10}{24} to lowest terms by extracting and canceling out 2.

$$y^{2}+\frac{5}{12}y=\frac{98}{24}$$

Reduce the fraction \frac{98}{24} to lowest terms by extracting and canceling out 2.

$$y^{2}+\frac{5}{12}y=\frac{49}{12}$$

Divide \frac{5}{12}, the coefficient of the x term, by 2 to get \frac{5}{24}. Then add the square of \frac{5}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{5}{12}y+\left(\frac{5}{24}\right)^{2}=\frac{49}{12}+\left(\frac{5}{24}\right)^{2}$$

Square \frac{5}{24} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{5}{12}y+\frac{25}{576}=\frac{49}{12}+\frac{25}{576}$$

Add \frac{49}{12} to \frac{25}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{5}{12}y+\frac{25}{576}=\frac{2377}{576}$$

Factor y^{2}+\frac{5}{12}y+\frac{25}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{5}{24}\right)^{2}=\frac{2377}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{5}{24}\right)^{2}}=\sqrt{\frac{2377}{576}}$$

Simplify.

$$y+\frac{5}{24}=\frac{\sqrt{2377}}{24} y+\frac{5}{24}=-\frac{\sqrt{2377}}{24}$$

Subtract \frac{5}{24} from both sides of the equation.

$$y=\frac{\sqrt{2377}-5}{24} y=\frac{-\sqrt{2377}-5}{24}$$