Question

Find the value of m so that the quadratic equation \[mx(x - 7) + 49 = 0\] has two equal roots.

Answer 1

Hint: Express the given equation in the standard form and find the coefficients. Then, find the discriminant \[D = {b^2} - 4ac\] and equate it to zero to find the value of m.

Complete step-by-step solution -
A quadratic equation is an equation with the degree or the highest power as two. It contains at least one term with the second power of the variable and it is the highest power in the expression.
The standard form of the quadratic equation is expressed as follows:
\[a{x^2} + bx + c = 0...........(1)\]
A polynomial has as many roots equal to its degree. Therefore, the quadratic equation has two roots which can be equal or unequal, real or complex.
The discriminant determines the nature of the roots. The discriminant is defined as follows:
\[D = {b^2} - 4ac..........(2)\]
If D > 0, then the roots are unequal and real.
If D < 0, then the roots are complex.
If D = 0, then the roots are real and equal.
We have the quadratic equation given as follows:
\[mx(x - 7) + 49 = 0\]
Let us write it in standard form as in equation (1).
\[m{x^2} - 7mx + 49 = 0..........(3)\]
Comparing, equation (3) with equation (1), we have:
\[a = m;b = - 7m;c = 49........(4)\]
Substituting equation (4) in equation (2) and equating it to zero for equal roots, we have:
\[{( - 7m)^2} - 4(m)(49) = 0\]
Simplifying the equation, we have:
\[49{m^2} - 4(49)m = 0\]
Dividing both sides of the equation by 49, we have:
\[{m^2} - 4m = 0\]
Taking m as a common term and solving, we get:
\[m(m - 4) = 0\]
\[m = 0;m = 4\]
We have two solutions. Let us examine the solution m = 0.
\[(0)x(x - 7) + 49 = 0\]
\[49 = 0\], which is never equal.
Hence, m = 0 is not possible.
Hence, the value of m is 4.

Note: You get two solutions for the value of m, do not conclude directly that m has two solutions. Substitute the solutions back in the equation to check if the conditions are satisfied.