The solution of a quadratic equation, cubic equation is possible with graph. I have found a method by which we can find the solution of a quadratic, cubic, etc, equations with the help of graph when one of the solution is known. The solution consists of finding a equation which relates the roots of the equation. It is very easy to find such a solution. Although the equation consists of only two roots but when we examine the roots then we can find all the roots of the equation. We will find the expression for quadratic equation in this post and discuss cubic equation in the next post. The method goes like this.
The two straight lines gives the solution of the equations x2 + 5x + c1 = 0 and
x2 − 5x + c2 = 0. If we know one root of the equation then we can find the other root by looking at the point's corresponding coordinate point where one coordinate is equal. For example as one roots of x2 + 5x + 6 = 0 is 2 so the corresponding point on the curve taking x = 2 is (2,3) so the other root is 3. We can also find the roots by looking at the point (a,b) where the area formed by the rectangle between x-axis, y-axis, x = a and y = b have area equal to the constant term. The condition for the roots for a particular equation is that the roots has product equal to c/a, where c is the constant term and a is coefficient of x2.
- Let the equation be ax2 + bx + c = 0
- Take two roots of the equation. Let it be x1 and x2.
- Form one equation with one root.
ax12 + bx1 + c = 0 ---(i)
Form one more equation with other root.
ax22 + bx2 + c = 0 ---(ii) - Subtract (ii) from (i) and solve it
(ax12 − ax22) + (bx1 − bx2) + c − c = 0
a(x12 − x22) + b(x1 − x2) = 0
(x1 − x2)[a(x1 + x2) + b] = 0
Suppose both the roots are different, then
[a(x1 + x2) + b] = 0
(x1 + x2) = −b/a You will get a relation in the roots. - Plot the relation on a graph by taking x1 to be y and x2 to be x.
The relation becomes y = −x − (b/a)
The two straight lines gives the solution of the equations x2 + 5x + c1 = 0 and
x2 − 5x + c2 = 0. If we know one root of the equation then we can find the other root by looking at the point's corresponding coordinate point where one coordinate is equal. For example as one roots of x2 + 5x + 6 = 0 is 2 so the corresponding point on the curve taking x = 2 is (2,3) so the other root is 3. We can also find the roots by looking at the point (a,b) where the area formed by the rectangle between x-axis, y-axis, x = a and y = b have area equal to the constant term. The condition for the roots for a particular equation is that the roots has product equal to c/a, where c is the constant term and a is coefficient of x2.
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