A quadratic expression when equated to zero is called a quadratic equation. A quadratic equation looks like this ax2 + bx + c = 0. In this post I will show you a method which is very good to find the solution of a quadratic equation. I found this method.Let us look at the method.
ax2 + bx + c = 0
We can write it as,
(x + b/a)x + c/a = 0
or (x + b/a)x = -c/a
Let A = x + b/a and B = x
Then,A-B = b/a, AB = -c/a and A+B = 2x + b/a
Applying the identity (A+B)2 = (A-B)2 + 4AB
(2x + b/a)2 = (b/a)2 - 4c/a = (b2 - 4ac)/a2
2x + b/a = ±(1/a)√(b2 - 4ac)
x = (-b ± √(b2 - 4ac))/2a
As we can see there are two values which satisfy the equation hence the number of solutions is two and there are two roots. As the solutions of a quadratic equation are called roots.
Let us analyze the roots i.e. when they are real. The value under the square root is positive if b2 - 4ac is positive. When such condition arises then the roots are real. The value b2 - 4ac is called the discriminant. If the discriminant is equal to zero then both the roots are equal. If the discriminant is negative then both the roots are imaginary and they occur in conjugate pairs. If the roots are real and distinct then the graph cuts the x-axis at two different points. If the roots are real and equal then the graph cuts the x-axis at one point. If the roots are imaginary then the graph does not cut the x-axis.
The graph below shows two real roots.
x2 + 5x - 2 = 0
The graph below shows two real roots
x2 + 5x + 6.25 = 0
The graph below represents when roots are imaginary.
ax2 + bx + c = 0
We can write it as,
(x + b/a)x + c/a = 0
or (x + b/a)x = -c/a
Let A = x + b/a and B = x
Then,A-B = b/a, AB = -c/a and A+B = 2x + b/a
Applying the identity (A+B)2 = (A-B)2 + 4AB
(2x + b/a)2 = (b/a)2 - 4c/a = (b2 - 4ac)/a2
2x + b/a = ±(1/a)√(b2 - 4ac)
x = (-b ± √(b2 - 4ac))/2a
As we can see there are two values which satisfy the equation hence the number of solutions is two and there are two roots. As the solutions of a quadratic equation are called roots.
Let us analyze the roots i.e. when they are real. The value under the square root is positive if b2 - 4ac is positive. When such condition arises then the roots are real. The value b2 - 4ac is called the discriminant. If the discriminant is equal to zero then both the roots are equal. If the discriminant is negative then both the roots are imaginary and they occur in conjugate pairs. If the roots are real and distinct then the graph cuts the x-axis at two different points. If the roots are real and equal then the graph cuts the x-axis at one point. If the roots are imaginary then the graph does not cut the x-axis.
The graph below shows two real roots.
x2 + 5x - 2 = 0
x2 + 5x + 6.25 = 0
x2 + 5x + 8 = 0
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