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Question: How do you solve \({x^2} + 9x = - 7\) graphically and algebraically?...

How do you solve x2+9x=−7{x^2} + 9x = - 7 graphically and algebraically?

Explanation

Solution

This equation is the quadratic equation. The general form of the quadratic equation is ax2+bx+c=0a{x^2} + bx + c = 0. Where ‘a’ is the coefficient of x2{x^2}, ‘b’ is the coefficient of x and ‘c’ is the constant term.
To solve this equation, we will apply the quadratic formula for the quadratic equation.
The formula is as below:
x=−b±b2−4ac2ax = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}.
Here, b2−4ac\sqrt {{b^2} - 4ac} is called the discriminant. And it is denoted by Δ\Delta .
If Δ\Delta is greater than 0, then we will get two distinct and real roots.
If Δ\Delta is less than 0, then we will not get real roots. In this case, we will get two complex numbers.
If Δ\Delta is equal to 0, then we will get two equal real roots.

Complete step by step solution:
Here, the quadratic equation is
⇒x2+9x=−7\Rightarrow {x^2} + 9x = - 7
Let us add 7 both sides.
⇒x2+9x+7=−7+7\Rightarrow {x^2} + 9x + 7 = - 7 + 7
That is equal to,
⇒x2+9x+7=0\Rightarrow {x^2} + 9x + 7 = 0
Let us compare the above expression with ax2+bx+c=0a{x^2} + bx + c = 0.
Here, we get the value of ‘a’ is 1, the value of ‘b’ is 9, and the value of ‘c’ is 7.
Now, let us find the discriminant Δ\Delta .
⇒Δ=b2−4ac\Rightarrow \Delta = {b^2} - 4ac
Let us substitute the values.
⇒Δ=(9)2−4(1)(7)\Rightarrow \Delta = {\left( 9 \right)^2} - 4\left( 1 \right)\left( 7 \right)
Simplify it.
⇒Δ=81−28\Rightarrow \Delta = 81 - 28
Subtract the right-hand side.
⇒Δ=53\Rightarrow \Delta = 53
Here, Δ\Delta is greater than 0, then we will get two different real roots.
Now,
x=−b±b2−4ac2ax = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}
Put all the values.
⇒x=−(9)±532(1)\Rightarrow x = \dfrac{{ - \left( 9 \right) \pm \sqrt {53} }}{{2\left( 1 \right)}}
That is equal to
⇒x=−9±532\Rightarrow x = \dfrac{{ - 9 \pm \sqrt {53} }}{2}
Hence, the two factors are −9+532\dfrac{{ - 9 + \sqrt {53} }}{2} and −9−532\dfrac{{ - 9 - \sqrt {53} }}{2}.
Solving the equation graphically would be inefficient since the algebraic method is required to use the graphical method. However, if you are given a graph with the exact value of the x-intercepts, the x-intercepts will be the solutions to the equation.

Note:
One important thing is, we can always check our work by multiplying out factors back together, and check that we have got back the original answer.
Here is a list of methods to solve quadratic equations:
Factorization
Completing the square
Using graph
Quadratic formula