Ax2 bx c what is a

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. A very common and easy-to-understand application is the height of a ball thrown at the ground off a building. Because gravity will make the ball speed up as it falls, a quadratic equation can be used to estimate its height any time before it hits the ground.

Note: The equation isn't completely accurate, because friction from the air will slow the ball down a little. For our purposes, this is close enough. A ball is thrown off a building from feet above the ground. The negative value means it's heading toward the ground. About how long does it take for the ball to hit the ground? When the ball hits the ground, the height is 0. Substitute 0 for h.

This equation is difficult to solve by factoring or by completing the square, so solve it by applying the Quadratic Formula,. In this case, the variable is t rather than x. Be very careful with the signs. Use a calculator to find both roots. And it's a " 2 a " under there, not just a plain " 2 ". Remember that " b 2 " means "the square of ALL of b , including its sign", so don't leave b 2 being negative, even if b is negative, because the square of a negative is a positive.

In other words, don't be sloppy and don't try to take shortcuts, because it will only hurt you in the long run. Trust me on this! This quadratic happens to factor, which I can use to confirm what I get from the Quadratic Formula. The Formula should give me the same answers. Now, what would my solution look like in the Quadratic Formula? 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:. If the equation factors we can find the points easily, but we may have to use the quadratic formula in some cases.

If the solutions are imaginary, that means that the parabola has no x -intercepts is strictly above or below the x -axis and never crosses it. If the solutions are real, but irrational radicals then we need to approximate their values and plot them. The y -intercept of any graph is a point on the y -axis and therefore has x -coordinate 0.