# Solving Quadratic Equations

A quadratic equation is of the form ax^{2} + bx + c = 0.
Solving an equation of this kind is

considerably more tricky than solving a linear equation . Fortunately, the
following method leads to

an easy formula that one could refer to when needed. Multiplying each term of
the equation by 4a,

4a^{2}x^{2} + 4abx + 4ac = 0. Adding and subtracting b^{2}, we get 4a^{2}x^{2} + 4abx +
b^{2} - b^{2} + 4ac = 0.

Transferring the last two terms of the LHS to the RHS, 4a^{2}x^{2} + 4abx + b^{2} =
b^{2} - 4ac. Note that

the LHS is now a perfect square of 2ax + b. Therefore, (2ax + b)^{2} = b^{2} - 4ac
which implies 2ax +

. This process has transformed the original quadratic equation into a linear
one!

Since we know how to solve linear equations from the previous section, we can
write the solution

to the current problem as

**Example
**Let us solve the equation x

^{2}= -10 * x - 9. It may be rewritten as x

^{2}+ 10x + 9 = 0. Noting that a

= 1, b = 10 and c = 9 for the given equation , we have Thus

there are 2 solutions to this equation : -1 and -9.

**Application
**

*Ming is considering a three-year investment in a development project. The project will yield*

positive returns of $1000 the first year for Ming, and negative returns of $100 and $1100 the

following two years. What bank rate of interest will make Ming indifferent between investing in the

project and not investing in it?

positive returns of $1000 the first year for Ming, and negative returns of $100 and $1100 the

following two years. What bank rate of interest will make Ming indifferent between investing in the

project and not investing in it?

To solve this problem, we need to figure out that rate of
interest which makes the present value of

the project equal to zero . (In economics, this special interest rate is called
the internal rate of

return of a project.) Thus 1000 - 100/(1 + r) - 1100/(1 + r)^{2} = 0. Multiplying
each term by (1 + r)

^{2}, we get 1000 * (1 + r)^{2} - 100 * (1 + r) - 1100 = 0. Collecting powers of r,

10 * r^{2} + 19 * r - 2 = 0. Using the formula for solving a quadratic equation,

which implies r is either -2 or 0.1. Neglecting the negative root , the

bank interest rate that makes Ming indifferent between investing in the project
or not is 10%.

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