Question

How do you solve \[12{x^2} = - 11x + 15\] by graphing?

Answer 1

Hint: In this question we are asked to solve the quadratic equation using graphical method, for this we will consider each term in the equation as a variable let it be \[y\] , i.e., \[y = 12{x^2}\] and \[y = - 11x + 15\] , and then we will plot each equation individually and find their point of intersection, and the \[x\] -coordinates of the point of intersection are the requires solutions for the given quadratic equation.

Complete step by step solution:
A graph is the picture of the points that make a function true. The roots of a quadratic equation are the \[x\] -intercepts of the graph. A graph is a useful representation for determining the solution of a quadratic equation. To best use a graph, think about a quadratic equation as written in two parts: \[a{x^2} + bx + c = k\] ,
Graph each part of the quadratic equation:  \[a{x^2} + bx + c = k\] and \[y = k\] , Look for the intersection of the two graphs. The \[x\] -coordinates of the intersection points will tell you the values of \[x\] that are solutions to the original equation,
Given equation is \[12{x^2} = - 11x + 15\] ,
Let us consider each term in the equation as a variable, i.e.,
 \[ \Rightarrow \] \[y = 12{x^2}\] and \[y = - 11x + 15\] ,
Now plot the graphs of these equations we get,

From the graph it is clear that the two equations \[y = 12{x^2}\] which is a curve and \[y = - 11x + 15\] which is a straight line intersect at points \[\left( { - 1.67,33.33} \right)\] and \[\left( {0.75,6.75} \right)\] ,
So, the solution for the given equation is \[x = - 1.67\] and \[x = 0.75\] .

The required solution for the given equation \[12{x^2} = - 11x + 15\] is equal to \[ - 1.67\] and \[0.75\] .

Note: It is important to note that the solution of the given equation is only \[x\] -coordinates of the point of intersection.
A quadratic equation has two roots if its graph has two \[x\] -intercepts
A quadratic equation has one root it its graph has one \[x\] -intercept
A quadratic equation has no real solutions if its graph has no \[x\] -intercepts.