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# How do you solve $7{x^2} - 5 = 2x + 9{x^2}$ using the quadratic formula?

Last updated date: 20th Jul 2024
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Hint: We first need to rearrange the given equation in the standard form of quadratic equation. That is in the form of $a{x^2} + bx + c = 0$. After that we can solve this using various methods that are by completing the square, factorization, graph or by quadratic formula. Here we need to use a quadratic formula that is $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.

Complete step-by-step solution:
Given, $7{x^2} - 5 = 2x + 9{x^2}$
Shifting the terms we have,
$2x + 9{x^2} - 7{x^2} + 5 = 0$
$2{x^2} - 2x + 5 = 0$
Since the degree of the equation is 2, we have 2 factors or two roots.
On comparing the given equation with the standard quadratic equation$a{x^2} + bx + c = 0$, we have$a = 2$, $b = - 2$ and $c = 5$.
Now we use quadratic formula or Sridhar’s formula,
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.
Substituting we have,
$\Rightarrow x = \dfrac{{ - ( - 2) \pm \sqrt {{{\left( { - 2} \right)}^2} - 4(2)(5)} }}{{2(2)}}$
$= \dfrac{{2 \pm \sqrt {4 - 40} }}{4}$
$= \dfrac{{2 \pm \sqrt { - 36} }}{4}$
$= \dfrac{{2 \pm \sqrt { - 1 \times 36} }}{4}$
We know that $\sqrt { - 1} = i$,
$= \dfrac{{2 \pm i\sqrt {36} }}{4}$
We know that 36 is a perfect square,
$= \dfrac{{2 \pm 6i}}{4}$
Taking 2 common,
$= \dfrac{{2(1 \pm 3i)}}{4}$
$= \dfrac{{1 \pm 3i}}{2}$
Thus we have two roots,
$\Rightarrow x = \dfrac{{1 + 3i}}{2}$ and $x = \dfrac{{1 - 3i}}{2}$. This is the required answer.

Note: Since we have a polynomial of degree two and hence it is called quadratic polynomial. If we have a polynomial of degree ‘n’ then we have ‘n’ roots. In the given problem we have a degree that is equal to 2. Hence the number of roots are 2. Also we know that $\sqrt { - 1}$ is undefined and we take $\sqrt { - 1} = i$ that is an imaginary number. Quadratic formula and Sridhar’s formula are both the same. We know that the product of two negative numbers gives us a positive number. Also keep in mind when shifting values from one side of the equation to another side of the equation, always change sign from positive to negative and vice-versa.