Real Solutions: Quadratic Equations Explained
Hey guys! Today, we're diving into the fascinating world of quadratic equations and figuring out how many real solutions each one has. We'll be sorting these equations into categories: those with no real solutions, those with exactly one, and those with two. Let's get started!
Understanding Real Solutions
Before we jump into the equations, let's quickly recap what real solutions are. When we solve a quadratic equation, we're essentially finding the values of x that make the equation true. These values are also the points where the graph of the quadratic equation (a parabola) intersects the x-axis. Real solutions are just those intersection points that are, well, real numbers! If the parabola never touches the x-axis, then there are no real solutions. If it touches at exactly one point, there's one real solution, and if it crosses at two points, there are two real solutions.
The Discriminant: Your Solution Detective
The key to figuring out how many real solutions a quadratic equation has without actually solving it is the discriminant. Remember the quadratic formula? It's:
x = (-b ± √(b² - 4ac)) / (2a)
The discriminant is the part under the square root: b² - 4ac. This little expression tells us everything we need to know:
- If b² - 4ac > 0: Two distinct real solutions
- If b² - 4ac = 0: Exactly one real solution (a repeated root)
- If b² - 4ac < 0: No real solutions (the solutions are complex numbers)
Analyzing the Equations
Okay, let's put on our detective hats and analyze each equation. We'll rewrite them in the standard quadratic form (ax² + bx + c = 0), identify a, b, and c, and then calculate the discriminant.
Equation 1: 5x² + 2 = 4x
First, let's rearrange the equation into the standard form:
5x² - 4x + 2 = 0
Now we can identify our coefficients:
- a = 5
- b = -4
- c = 2
Let's calculate the discriminant:
b² - 4ac = (-4)² - 4 * 5 * 2 = 16 - 40 = -24
Since the discriminant is negative (-24 < 0), this equation has no real solutions. The parabola represented by this equation never intersects the x-axis. Graphically, it floats either entirely above or entirely below the x-axis.
Think about it like this: you're trying to find a number that, when plugged into the equation, makes it equal to zero. But the smallest value that 5x² - 4x + 2 can take is greater than zero. This is because the positive x² term always dominates, and the constant term is also positive. So, no matter what real number you substitute for x, you'll never get zero. The lack of real solutions makes this equation somewhat unique.
Equation 2: 4x² - 16x = 0
This equation is already in a pretty good form. We can identify the coefficients as:
- a = 4
- b = -16
- c = 0
Calculate the discriminant:
b² - 4ac = (-16)² - 4 * 4 * 0 = 256 - 0 = 256
Since the discriminant is positive (256 > 0), this equation has two real solutions. This means the parabola intersects the x-axis at two distinct points. We can even find these points by factoring the equation:
4x² - 16x = 0 4x(x - 4) = 0
So, x = 0 or x = 4. These are our two real solutions!
The fact that the constant term (c) is zero means that one of the solutions will always be zero. This also tells us that the parabola passes through the origin (0,0). The other solution, x=4, indicates the second point where the parabola intersects the x-axis. These two solutions give us a good picture of how this parabola behaves on the graph.
Equation 3: 3(x + 5)² = -2
Let's rewrite this equation in standard form. First, expand the squared term:
3(x² + 10x + 25) = -2 3x² + 30x + 75 = -2 3x² + 30x + 77 = 0
Now we can identify a, b, and c:
- a = 3
- b = 30
- c = 77
Calculate the discriminant:
b² - 4ac = (30)² - 4 * 3 * 77 = 900 - 924 = -24
Since the discriminant is negative (-24 < 0), this equation has no real solutions. This might seem a bit counterintuitive at first because of the squared term. However, notice that the original equation 3(x + 5)² = -2 implies that a squared number is equal to a negative number, which is impossible for any real value of x. The squared term will always result in a non-negative number, and when multiplied by a positive constant (3), it can never equal a negative number. This quick observation could save time without having to expand the equation and calculating the discriminant.
Equation 4: 3x² + 24x = -48
Let's get this equation into standard form:
3x² + 24x + 48 = 0
Identify the coefficients:
- a = 3
- b = 24
- c = 48
Calculate the discriminant:
b² - 4ac = (24)² - 4 * 3 * 48 = 576 - 576 = 0
Since the discriminant is zero, this equation has exactly one real solution. This means the parabola touches the x-axis at only one point. We can find this solution by factoring or using the quadratic formula. Let's simplify the equation first by dividing by 3:
x² + 8x + 16 = 0
Now, factor:
(x + 4)² = 0
So, x = -4. This is our repeated root, the single point where the parabola touches the x-axis.
Because the discriminant is zero, the quadratic formula will yield one solution. The two parts of the ± term vanish, giving only one value for x. This also means that the vertex of the parabola lies exactly on the x-axis. In this case, the vertex is at the point (-4, 0). This equation represents a special case where the parabola just touches the x-axis at one point, rather than crossing it at two.
Categorizing the Equations
Alright, time to put our findings into the table:
| No Real Solutions | Exactly One Real Solution | Two Real Solutions |
|---|---|---|
| 5x² + 2 = 4x | 3x² + 24x = -48 | 4x² - 16x = 0 |
| 3(x + 5)² = -2 |
Conclusion
And there you have it! We successfully determined the number of real solutions for each quadratic equation using the discriminant and categorized them accordingly. Remember, the discriminant is your friend when you need to quickly figure out how many real solutions a quadratic equation has. Keep practicing, and you'll become a quadratic equation pro in no time!