What is a quadratic function? I’m having some trouble memorizing the formula to find the roots
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A quadratic function is a polynomial of degree 2, meaning the highest power of x is 2. It has the general form f of x equals a x squared plus b x plus c, where a cannot be zero. The graph of a quadratic function is always a parabola. When a is positive, the parabola opens upward, and when a is negative, it opens downward. The vertex is the highest or lowest point of the parabola, and the axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror images.
The roots of a quadratic function are the x-values where the parabola intersects the x-axis, which means where y equals zero. There are three possible cases. First, a parabola can cross the x-axis at two points, giving us two real roots. Second, a parabola can touch the x-axis at exactly one point, called the vertex, giving us one real root with multiplicity two. Third, a parabola may not touch the x-axis at all, meaning there are no real roots. Understanding these visual patterns helps us predict what we'll find when we solve quadratic equations algebraically.
Let's derive the quadratic formula step by step using the completing the square method. We start with the general form a x squared plus b x plus c equals zero. First, we move the constant term c to the right side. Next, we divide everything by a to get x squared plus b over a times x equals negative c over a. Now comes the key step: completing the square. We add the square of half the coefficient of x to both sides. This gives us b over 2a quantity squared. The left side becomes a perfect square: x plus b over 2a quantity squared. After simplifying the right side, we get b squared minus 4ac over 4a squared. Taking the square root of both sides introduces the plus or minus symbol. Finally, we solve for x by subtracting b over 2a from both sides, giving us the quadratic formula: x equals negative b plus or minus the square root of b squared minus 4ac, all over 2a.
Let's break down each component of the quadratic formula to make it easier to remember and understand. The formula has four main parts. First, negative b - we always take the negative of the coefficient of x. Second, the plus or minus symbol indicates that we get two possible solutions for most quadratic equations. Third, we have the discriminant under the square root: b squared minus 4ac. This discriminant is crucial because it tells us how many real roots we have. If it's positive, we get two real roots. If it's zero, we get one real root. If it's negative, we get no real roots. Finally, everything is divided by 2a, which is twice the coefficient of x squared. A helpful memory aid is to say: negative b, plus or minus the square root, over 2a. This rhythm helps many students remember the formula structure.
Let's work through three examples to see how the quadratic formula works in practice. Example 1: x squared minus 5x plus 6 equals zero. Here, a equals 1, b equals negative 5, and c equals 6. The discriminant is b squared minus 4ac, which is 25 minus 24, equals 1. Since this is positive, we have two real roots. Substituting into the formula: x equals 5 plus or minus 1, all over 2, giving us x equals 3 or x equals 2. On the graph, we can see the parabola crosses the x-axis at these two points. Example 2: x squared minus 4x plus 4 equals zero. Here a equals 1, b equals negative 4, c equals 4. The discriminant is 16 minus 16, which equals zero. This means one real root with multiplicity 2. The solution is x equals 4 over 2, which equals 2. The parabola touches the x-axis at exactly one point. Example 3: x squared plus x plus 1 equals zero. Here a equals 1, b equals 1, c equals 1. The discriminant is 1 minus 4, which equals negative 3. Since this is negative, there are no real solutions. The parabola doesn't intersect the x-axis at all.