Question

How do you solve using the quadratic formula $\left( {3 - y} \right)\left( {y + 4} \right) = 3y - 5?$

Answer 1

Hint: In this question, we are going to solve the given equation by using the quadratic formula and then find the value of $y$.
First, we have to expand the given term on the left hand side of the equation.
Next we are going to solve the equation and write it in the quadratic form.
Then we are going to solve the quadratic equation by using the quadratic formula.
Hence we can get the required result.

Formula used: The quadratic formula can be written as
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Here ${b^2} - 4ac$ is the formula for discriminant.
There are three possible outcomes for the discriminant.
${b^2} - 4ac < 0$, the quadratic equation has no real solutions.
${b^2} - 4ac = 0,$ The quadratic equation has only one solution or two real and equal solutions.
${b^2} - 4ac > 0,$ The quadratic equation has two real and distinct solutions.

Complete step-by-step solution:
In this question we are going to solve the given equation by using the quadratic formula.
First write the given equation $\left( {3 - y} \right)\left( {y + 4} \right) = 3y - 5$
Now we are going to expand the term on the left hand side of the equation.
$ \Rightarrow 3y + 12 - {y^2} - 4y = 3y - 5$
Combining and adding the like terms we get,
$ \Rightarrow 12 - {y^2} - y = 3y - 5$
Bringing the right hand side terms to the left hand side we get,
$ \Rightarrow - {y^2} - y - 3y + 5 + 12 = 0$
$ \Rightarrow - {y^2} - 4y + 17 = 0$
Multiply the equation by minus we get,
$ \Rightarrow {y^2} + 4y - 17 = 0$
Therefore the above equation is of the quadratic form,
Here $a = 1,\,b = 4,\,c = - 17$
Applying those values to the quadratic formula we get,
$ \Rightarrow y = \dfrac{{ - 4 \pm \sqrt {{{\left( 4 \right)}^2} - 4\left( 1 \right)\left( { - 17} \right)} }}{{2\left( 1 \right)}}$
On simplify the term and we get,
$ \Rightarrow y = \dfrac{{ - 4 \pm \sqrt {16 + 68} }}{2}$
Let us add the term and we get,
$ \Rightarrow y = \dfrac{{ - 4 \pm \sqrt {84} }}{2}$
On splitting the term and we get
$ \Rightarrow y = \dfrac{{ - 4 \pm \sqrt {4 \times 21} }}{2}$
On squaring the term and we get
$ \Rightarrow y = \dfrac{{ - 4 \pm 2\sqrt {21} }}{2}$
Taking 2 as common we get
$ \Rightarrow y = \dfrac{{2( - 2 \pm \sqrt {21} )}}{2}$
On cancel the term and we get
$ \Rightarrow y = - 2 \pm \sqrt {21} $
Taking the square value out
$ \Rightarrow y = - 2 \pm 4.58$
On splitting the term and we get
$ \Rightarrow y = - 2 + 4.58,y = - 2 - 4.58$
On simplify we get
$ \Rightarrow y = 2.58,y = - 6.58$

Thus the value of $y$ is $ - 6.58,2.58$

Note: There are various methods to solve the quadratic equation. Some of them are as follows: factoring, using the square roots, completing the square and the quadratic formula.
Among the four types of solving the equation, using a quadratic formula is more appropriate and can provide the exact amount value of unknown variables, because not all quadratic equations can be solved using the three methods, but all quadratic equations can be solved using the quadratic formula.