Question

How do you solve the equation ${x^2} + 2x - 35?$

Answer 1

Hint: In this question, we are going to solve the given equation and then find the value of $x.$
In this we are going to factor the expression by splitting the middle term.
Multiply the coefficient of the first term by a constant in the given expression, we get a number and then find two factors for that number whose sum equals the coefficient of the middle term.
Now rewrite the polynomial by splitting the middle term using the two factors that found before and then add the first two terms and last two terms, taking common factors outside from the first and last two terms. Add the four terms of the above step and we get the desired factorization.
And then solve the factorization to get the values of $x.$
Hence we get the required result.

Complete step-by-step solution:
First write the given expression and mark it as$\left( 1 \right)$.
$ \Rightarrow {x^2} + 2x - 35...\left( 1 \right)$
The given expression is of the quadratic form $a{x^2} + bx + c = 0$
Here the first term is ${x^2}$ and its coefficient is $1.$
The middle term is $2x$ and its coefficient is $2.$
The last term is $ - 35$ and it is a constant.
First we are going to multiply the coefficient of the first term by the last term.
That is, $1 \times - 35 = - 35$
Next we are going to find factors of $ - 35$ whose sum is equal to $2.$
$ \Rightarrow 7 - 5 = 2$
By splitting the middle term using the factors $7$ and $ - 5$in the given expression.
$ \Rightarrow {x^2} + 7x - 5x - 35$
Taking common factors outside from the two pairs
$ \Rightarrow x\left( {x + 7} \right) - 5(x + 7)$
Rewrite in the factored form,
$ \Rightarrow \left( {x - 5} \right)\left( {x + 7} \right)$
$ \Rightarrow \left( {x - 5} \right) = 0,\left( {x + 7} \right) = 0$
$ \Rightarrow x = 5,x = - 7$
$ \Rightarrow x = 5, - 7$
The required factors of the expression ${x^2} + 2x - 35$ are $\left( {x - 5} \right)\left( {x + 7} \right)$ and

The values of $x$ is $5$ and $ - 7$.

Note: We can check our factoring by multiplying them all out to see if we get the original expression. If we do, our factoring is correct, otherwise we had to try again.
We can also solve this equation by using the quadratic formula.
The given equation is of a quadratic form
${x^2} + 2x - 35$
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Here $a = 1,\,b = 2,\,c = - 35$
$ \Rightarrow x = \dfrac{{ - 2 \pm \sqrt {{{\left( 2 \right)}^2} - 4\left( 1 \right)\left( { - 35} \right)} }}{{2\left( 1 \right)}}$
On simplify the numerator term and we get,
$ \Rightarrow x = \dfrac{{ - 2 \pm \sqrt {4 + 140} }}{2}$
On add the term and we get,
$ \Rightarrow x = \dfrac{{ - 2 \pm \sqrt {144} }}{2}$
On taking square root we get
$ \Rightarrow x = \dfrac{{ - 2 \pm 12}}{2}$
Now we have split the term and we get,
$ \Rightarrow x = \dfrac{{ - 2 + 12}}{2}$
$ \Rightarrow x = \dfrac{{10}}{2}$
$\therefore x = 5$
Also,
$ \Rightarrow x = \dfrac{{ - 2 - 12}}{2}$
$ \Rightarrow x = \dfrac{{ - 14}}{2}$
$\therefore x = - 7$
Hence we get the required values.