Question

How do you solve $4{x^2} + 7x - 17 = 3{x^2} + 12x - 3$?

Answer 1

Hint: This problem deals with solving 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 answer:
The given equation is $4{x^2} + 7x - 17 = 3{x^2} + 12x - 3$, consider it as given below:
$ \Rightarrow 4{x^2} + 7x - 17 = 3{x^2} + 12x - 3$
Now group all the like terms like ${x^2}$ terms, $x$ terms and constants as shown below:
$ \Rightarrow 4{x^2} - 3{x^2} + 7x - 12x - 17 + 3 = 0$
Now simplify the above equation, as given below:
$ \Rightarrow {x^2} - 5x - 14 = 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 ${x^2} - 5x - 14 = 0$ with the standard form $a{x^2} + bx + c = 0$ and compare the coefficients $a,b$ and $c$:
$ \Rightarrow a = 1,b = - 5$ and $c = - 14$
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 $a,b$ and $c$ in the above formula:
$ \Rightarrow x = \dfrac{{ - \left( { - 5} \right) \pm \sqrt {{{\left( { - 5} \right)}^2} - 4\left( 1 \right)\left( { - 14} \right)} }}{{2(1)}}$
$ \Rightarrow x = \dfrac{{5 \pm \sqrt {25 + 56} }}{2}$
Simplifying the above expression, as given below:
$ \Rightarrow x = \dfrac{{5 \pm \sqrt {81} }}{2}$
We know that the square root of 81 is 9, $\sqrt {81} = 9$
$ \Rightarrow x = \dfrac{{5 \pm 9}}{2}$
Now considering the two cases, with plus and minus, as shown:
$ \Rightarrow x = \dfrac{{5 + 9}}{2};x = \dfrac{{5 - 9}}{2}$
$ \Rightarrow x = \dfrac{{14}}{2};x = \dfrac{{ - 4}}{2}$
Hence the value of the roots are equal to :
$\therefore x = 7;x = - 2$

Final Answer: The solution of the equation $4{x^2} + 7x - 17 = 3{x^2} + 12x - 3$ are 7 and -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 on the right side. But here we are adding and subtracting some terms in order to factor.