# GMAT Algebra – Solving Quadratic Equations (Refresher)

Let’s nail down the quadratic formula in this video! I did my job. Now it’s your turn, tree.
Get growing. A quadratic equation refers to an equation
with the second degree of power. Remember that a linear equation is
an equation with the first degree of power. The quadratic equation has
its variable with the power of 2! For example, x² + 2 equals 0 is the quadratic equation, but x+2 equals 0 is the linear equation. With plenty of practices you can solve
quadratic equations by factoring. However in some cases, you might be frustrated to find that at first glance, it won’t be an easy task to solve it. That’s when we use the quadratic formula! The quadratic formula looks very intimidating. So if you are taking the GMAT soon, say within 1 month, forget about this section and skip the quadratic formula. It’s better to sharpen other math and verbal skills. If you have more than two months to prepare for the GMAT, go for it. Anyways, let us look closely at the quadratic formula. For any quadratic formula,
you can solve for x with this formula. If you have ax² +bx + c=0, and “a” is not equal to 0, and if you know the coefficients “a” and “b”,
and the constant “c”, you can solve for x. The quadratic formula is x equals minus “b”, plus or minus square root of “b” square minus 4 “ac”, divided by 2a. Notice that x has two values
because we are taking the value of plus or minus. Let us look at an example. Solve x² + 3x + 2 equals 0 using the quadratic formula. Here, “a” is equal to 1, “b” is equal to 3,
and “c” is equal to 2. Just plug in the value into the quadratic formula. you get x equals -3 plus or minus square root of
3 square minus 4 times 1 times 2, divided by 2 times 1. Simplify the terms in the square root. Then you have x equals minus 3, plus or minus square root of 9 minus 8, divided by 2. We can simplify 9 minus 8 inside the square root,
and it becomes square root of 1. Square root of 1 is equal to 1, so we have x equals minus 3 plus or minus 1, divided by 2. So we have two values of x here. For one value, we have minus 3 plus 1 divided by 2, which is equal to -1. For another value, we have minus 3 minus 1 divided by 2, which is equal to -2. For final answer, we can say that x has
either the value of -1 or -2. There are special cases for the quadratic formula. Usually we will have 2 values of x
when using the formula. However, what if we have identical values of x,
which means only 1 solution? For example, if the terms in square root in the formula
is equal to 0, then we only have 1 value of x. The plus or minus sign in the middle
does not matter any more, because adding or subtracting 0
does not change anything. If you plot this in a coordinate geometry, the quadratic equation of x
usually involves a curved parabolic shape and when the end tip of the parabola
meets exactly the x-axis, it will have only one value of x as solution. Another case is when the terms in the square root
of the formula has negative value. If you have a negative value in a square root,
it is not solvable in real number. It is the imaginary number,
but we don’t have to know that for the GMAT. For example, if we have sqaure root of -2,
there’s no solution for x. If you plot this in coordinate geometry,
the parabola does not touch the x-axis at all! Thus we do not have any solutions for x. In conclusion, just remember that
normally you have 2 solutions when solving using the quadratic formula. Be sure to double check the plus and minus sign! If you get square root of 0,
you only have 1 solution for x. If you get square root of negative number,
you don’t have any solutions. Now it’s the time for the GMAT level problem! Don’t forget the scrap papers and a pen;
solve for yourself. [reading the question] We know that the area of a rectangle is
the length times the width. And we also know the value of area is 3000. Writing a rectangular area equation, it is
x times x minus 130 equals 3000. Expanding it, we get x² – 130x equals 3000. Now move the constant 3000 so that
we have an equation equal to 0. Subtract 3000 from both sides. Then we have x² – 130x -3000 equals 0. Now, let’s use the quadratic formula! “a” is 1, “b” is -130, and “c” is -3000. Plug in these three values into the quadratic formula. x equals minus negative 130,
plus or minus square root of negative 130 square minus 4 times 1 times negative 3000, divided by 2 times 1. Simplify the equation. x equals 130 plus or minus,
square root of 16900 plus 12000, divided by 2. The constant in the square root
can be added together to 28900. Now we wonder, what is square root of 28900? The number looks HUGE! In this case, approximation
helps us to find the answer quickly. From the number square root of 28900, what is the closest number that can easily get the root? Let’s try 40,000. We know that 200 times 200 is 40,000. So square root of 28900 has to be smaller than 200! In the GMAT, many of the square root number
don’t stay as the radicals, or root numbers. So let us look at the number square root of 28900,
which has to be smaller than 200 and think about the hundreds digit 9. What two same number would
multiply together and produce the end digit 9? Maybe seven? 7 times 7 is 49! So let us try 17 times 17. It’s 289! Voila! So square root of 28900 has to be 170. Now that we found the square root,
let’s continue the calculation. We have x equals 130 plus or minus 170, divided by 2. Here, x has 2 possibilities. It’s either 130 plus 170, divided by 2 OR, 130 minus 170, divided by 2. But remember, we are dealing with
the length of a rectangle! The length of a rectangle CANNOT be a negative value. So disregard the 130 minus 170 divided by 2, because it is a negative number. With only one x value possible, 130 plus 170 divided by 2 is 150 and the answer is C. Check out cognifygmat.com and the link below for free quadratic formula problems! Have a great studying!