Question

how to solve the equation 6(y-3)^2-(y-3)-15=0​

Answer 1

Solution!!

→ 6(y - 3)² - (y - 3) - 15 = 0

Expand the expression using (a - b)² = a² + b² - 2ab.

→ 6((y)² + (3)² - 2(y)(3)) - (y - 3) - 15 = 0

Calculate the exponents.

→ 6(y² + 9 - 6y) - (y - 3) - 15 = 0

Distribute 6 through the parentheses.

→ 6y² + 54 - 36y - (y - 3) - 15 = 0

When there is a - sign in front of an expression in parentheses, change the sign of each term in the expression.

→ 6y² + 54 - 36y - y + 3 - 15 = 0

Collect like term.

→ 6y² - 36y - y + 54 + 3 - 15 = 0

Calculate the sum and the difference.

→ 6y² - 37y + 42 = 0

This is similar to the quadratic equation ax² + bx + c = 0 where a is 6, b is - 37, c is 42 and x is y. So, let's use the quadratic formula to calculate the value of y.

→ $\sf x=\dfrac{-b\pm \sqrt{b^{2}-4ac}}{2a}$

Substitute the values.

→ $\sf y=\dfrac{-(-37)\pm \sqrt{(-37)^{2}-4\times 6\times 42}}{2(6)}$

When there is a - in front of an expression in parentheses, change the sign of each term in the expression.

→ $\sf y=\dfrac{37\pm \sqrt{(-37)^{2}-4\times 6\times 42}}{2(6)}$

Calculate the product.

→ $\sf y=\dfrac{37\pm \sqrt{(-37)^{2}-1008}}{12}$

Evaluate the power.

→ $\sf y=\dfrac{37\pm \sqrt{1369-1008}{12}$

Subtract the values.

→ $\sf y=\dfrac{37\pm \sqrt{361}}{12}$

Calculate the square root.

→ $\sf y=\dfrac{37\pm 19}{12}$

Write two solutions, one with a + sign and the other with a - sign.

→ $\sf y=\dfrac{37+19}{12}$

→ $\sf y=\dfrac{37-19}{12}$

Calculate the value and simplify the expressions.

→ $\sf y=\dfrac{14}{3}$

→ $\sf y=\dfrac{3}{2}$

Hence, we have 2 values of y.