Ax2 bx c what is x

Note that the subtraction sign means the constant c is negative. Substitute the values into the Quadratic Formula. Simplify, being careful to get the signs correct. Simplify the radical:. Separate and simplify to find the solutions to the quadratic equation.

Note that in one, 6 is added and in the other, 6 is subtracted. The power of the Quadratic Formula is that it can be used to solve any quadratic equation, even those where finding number combinations will not work. Most of the quadratic equations you've looked at have two solutions, like the one above. The following example is a little different.

Subtract 6 x from each side and add 16 to both sides to put the equation in standard form. Identify the coefficients a , b , and c. Since 8 x is subtracted, b is negative. Since the square root of 0 is 0, and both adding and subtracting 0 give the same result, there is only one possible value.

Again, check using the original equation. Let's try one final example. This one also has a difference in the solution. Simplify the radical, but notice that the number under the radical symbol is negative! Check these solutions in the original equation. Be careful when expanding the squares and replacing i 2 with You may have incorrectly factored the left side as x — 2 2.

The correct answer is or. Using the formula,. If you forget that the denominator is under both terms in the numerator, you might get or. However, the correct simplification is , so the answer is or. The Discriminant. These examples have shown that a quadratic equation may have two real solutions, one real solution, or two complex solutions.

In the Quadratic Formula, the expression underneath the radical symbol determines the number and type of solutions the formula will reveal. You can always find the square root of a positive, so evaluating the Quadratic Formula will result in two real solutions one by adding the positive square root, and one by subtracting it. There will be one real solution. Since you cannot find the square root of a negative number using real numbers, there are no real solutions.

However, you can use imaginary numbers. You will then have two complex solutions, one by adding the imaginary square root and one by subtracting it. Use the discriminant to determine how many and what kind of solutions the quadratic equation. Evaluate b 2 — 4 ac. The result is a negative number. The discriminant is negative, so the quadratic equation has two complex solutions. Suppose a quadratic equation has a discriminant that evaluates to zero. Which of the following statements is always true?

A The equation has two solutions. B The equation has one solution. C The equation has zero solutions. A discriminant of zero means the equation has one solution. When the discriminant is zero, the equation will have one solution. Applying the Quadratic Formula. Quadratic equations are widely used in science, business, and engineering.

Quadratic equations are commonly used in situations where two things are multiplied together and they both depend on the same variable. For example, when working with area, if both dimensions are written in terms of the same variable, you use a quadratic equation. Because the quantity of a product sold often depends on the price, you sometimes use a quadratic equation to represent revenue as a product of the price and the quantity sold. Quadratic equations are also used when gravity is involved, such as the path of a ball or the shape of cables in a suspension bridge.

We can use this fact to find the y -intercepts by simply plugging 0 for x in the original equation and simplifying. So the y -intercept of any parabola is always at 0,c. To find the x -coordinate for the vertex we use the following formula:. Since "a" is positive we'll have a parabola that opens upward is U shaped. For this particular quadratic equation, factoring would probably be the faster method.

But the Quadratic Formula is a plug-n-chug method that will always work. Having "brain freeze" on a test and can't factor worth a darn? Use the plug-n-chug Formula; it'll always take care of you! The x -intercepts of the graph of a quadratic are the points where the parabola crosses the x -axis. This means that there must then be two x -intercepts on the graph.

Graphing, we get the curve below:. This can be useful if you have a graphing calculator, because you can use the Quadratic Formula when necessary to solve a quadratic, and then use your graphing calculator to make sure that the displayed x -intercepts have the same decimal values as do the solutions provided by the Quadratic Formula.

Note, however, that the calculator's display of the graph will probably have some pixel-related round-off error, so you'd be checking to see that the computed and graphed values were reasonably close; don't expect an exact match. I will apply the Quadratic Formula. In general, no, you really shouldn't. The "solution" or "roots" or "zeroes" of a quadratic are usually required to be in the "exact" form of the answer.

You can use the rounded form when graphing if necessary , but "the answer s " from the Quadratic Formula should be written out in the often messy "exact" form. In the example above, the exact form is the one with the square roots of ten in it.