Question

What are the solutions of $4{{x}^{2}}-7x=3x+24$ ?

Answer 1

Hint: From the question we have been asked to find the solutions of a quadratic equation. Here, given a quadratic equation expression, we have to simplify the expression and make it into a standard form of quadratic equation. If the quadratic equation is in the form of $a{{x}^{2}}+bx+c=0$, then we know that the roots of this quadratic equation are given by
$\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

Complete step-by-step solution:
So, from the question we have that,
$\Rightarrow 4{{x}^{2}}-7x=3x+24$
Now we will group all the like terms like ${{x}^{2}}$ terms x terms and constants as shown below:
$\Rightarrow 4{{x}^{2}}-7x-3x-24=0$
Now we will simplify the above equation, so the equation will be as given below,
$\Rightarrow 4{{x}^{2}}-10x-24=0$
Now the above equation is in the standard form of a quadratic equation, which is $a{{x}^{2}}+bx+c=0$
Here comparing the equation $4{{x}^{2}}-10x-24=0$ with the standard form $a{{x}^{2}}+bx+c=0$ and compare the coefficients a, b and c.
$\Rightarrow a=4,b=-10,c=-24$
Now applying the formula to find the value of the roots of x, as given below
$\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Substituting the values of a band c in the above formula
$\Rightarrow x=\dfrac{-\left( -10 \right)\pm \sqrt{{{\left( -10 \right)}^{2}}-4\times 4\times -24}}{2\times 4}$
Simplifying the above expression, as given below
$\Rightarrow x=\dfrac{10\pm \sqrt{100+384}}{8}$
 $\Rightarrow x=\dfrac{10\pm 22}{8}$
Now considering the two cases, with plus and minus, as shown
$\Rightarrow x=\dfrac{10+22}{8};x=\dfrac{10-22}{8}$
$\Rightarrow x=\dfrac{32}{8};x=\dfrac{-12}{8}$
$\Rightarrow x=4;x=\dfrac{-3}{2}$
Hence the value of the roots or the solutions are equal to
$\Rightarrow x=4;x=\dfrac{-3}{2}$

Note: Please note that this problem can also be done either by the method of completing the square or just factoring and solving the quadratic equation to solve $a{{x}^{2}}+bx+c=0$ by completing the square transform the equation so that the constant term c is alone or the right side. But here we are adding and subtracting some terms in order to factor.