Quadratic Equation Show
In algebra, a quadratic equation (from Latin quadratus 'square') is any equation that can be rearranged in standard form as where x represents an unknown, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 (and b ≠ 0) then the equation is linear, not quadratic, as there is no ax² term.) The numbers a, b, and c are the coefficients of the equation and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term. We can help you solve an equation of the form "ax2 + bx + c = 0" algebra/images/quadratic-solver.js Is it Quadratic?Only if it can be put in the form ax2 + bx + c = 0, and a is not zero. The name comes from "quad" meaning square, as the variable is squared (in other words x2). These are all quadratic equations in disguise:
How Does this Work?The solution(s) to a quadratic equation can be calculated using the Quadratic Formula:  The "±" means we need to do a plus AND a minus, so there are normally TWO solutions ! The blue part (b2 - 4ac) is called the "discriminant", because it can "discriminate" between the possible types of answer:
Learn more at Quadratic Equations Note: you can still access the old version here. Explore Quadratic equations Picture it: You’re at the kitchen table, with a convoluted quadratic equation in front of you. You don’t want to solve it. You don’t even need to solve it. All you actually need to find is the number of solutions. Is that even possible? You’ll be happy to know that YES, it is! That’s right — we can find just the number of solutions without having to find the solutions themselves! (cue happy dance) But how do we find the number of solutions? We use the discriminant! What is the discriminant?The discriminant of a quadratic equation written in standard form $$ax^{2}+bx+c=0$$ is the value of the expression: $$b^{2}-4ac$$ This is the value that determines the number of solutions to a quadratic equation — so basically, it’s our new best friend. How many real solutions can a quadratic equation have?Quadratic equations can have $$0$$, $$1$$, or $$2$$ solutions. The number of real solutions of a quadratic equation depends on the sign of the discriminant $$b^2-4ac$$ of that quadratic equation.
Why is the number of solutions useful?Quite frankly, we don’t always need the exact values of solutions; sometimes, we just need to know how many there are. For example, we may want to know if the related graph intersects the $$x$$-axis and, if it does, at how many points. We get that information from the number of solutions of a quadratic equation! How to find the number of solutions to a quadratic equationOkay, so you’ve found yourself in a situation where you only need to know the number of solutions — but you’re not sure where to start. Luckily, we do! Let’s walk through an example together. ExampleOur goal is to find the number of solutions to this quadratic equation: $$9x^{2} + 42x + 49 = 0$$ Our first order of business is to identify the coefficients $$a$$, $$b$$, and $$c$$ (this helps us calculate the discriminant): $$a=9, b=42, c=49$$ We’ve identified the coefficients, so we can evaluate the discriminant by substituting these coefficients into the expression $$b^2-4ac$$: $$42^2-4\times9\times49$$ Now, we’ll evaluate the power (remember your PEMDAS!): $$1764-4\times9\times49$$ Next is calculating the product: $$1764-1764$$ The sum of two opposites equals to $$0$$, so: $$0$$ The discriminant equals $$0$$, which means the quadratic equation has one real solution! $$1\text{ real solution}$$ Not so bad once you see it in context! Let’s summarize the steps so you can apply it beyond this example: Study summary
Do it yourself!If you’ve been around here awhile, you know we’re big advocates of putting your learnings into practice. Once you get the hang of it, it won’t take long to make these calculations — but step one is to truly get the hang of it! That’s where these problems come in. Find the number of solutions to the following quadratic equations:
Solutions:
How did that practice feel? Do you need some additional guidance? Try scanning a practice problem (or any other problem!) with your Photomath app so we can walk through each step. Here’s a sneak peek of what you’ll see: Extra creditHow to visualize the relation of the discriminant and the number of solutions: Want to go a step deeper? If you’re a visual learner, this will help you understand why the discriminant tells us the number of solutions! The graph of a quadratic function is a parabola, or a sort of U-shaped curve. When placed in an $$xy$$- coordinate system, it faces upwards or downwards. When we look for the number of solutions to a quadratic equation written in standard form $$ax^2+bx+c=0$$, we’re also looking for the number of $$x$$-intercepts of the related parabola. That parabola can intersect the $$x$$-axis in zero, one or two points:
Take any quadratic equation you want! If you follow these steps, you’ll notice the same things:
Pretty cool when it clicks, right? What is the formula of real solution?The solution of a quadratic equation ax2 + bx + c = 0 is given by the quadratic formula x = [-b ± √(b2 - 4ac)] / 2a, to find the solution of a quadratic equation. In the case of one real solution, the value of discriminant b2 - 4ac is zero. For example, x2 + 2x + 1 = 0 has only one solution x = -1.
What does number of real solutions mean?In terms of real solutions, there are always either 0, 1, or 2 real solutions to a quadratic equation, depending on the sign of the discriminant. If the discriminant is positive, there are exactly 2 real solutions. If the discriminant is zero, there is exactly one real solution (a repeated root).
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