When Does a Quadratic Have One Solution?
Quadratic equations are an essential topic in algebra that students encounter during their academic journey. A quadratic equation is a polynomial equation of the second degree, represented by the standard form: ax^2 + bx + c = 0, where a, b, and c are constants, and x represents the unknown variable. One of the primary objectives when solving a quadratic equation is to determine the number of solutions it possesses. In some cases, a quadratic equation has only one solution. This article will explore the conditions under which this occurs and provide an in-depth analysis of the topic.
Determining the number of solutions of a quadratic equation is closely linked to the concept of discriminant. The discriminant of a quadratic equation is the expression under the square root sign in the quadratic formula, Δ = b^2 – 4ac. This discriminant plays a crucial role in identifying the nature and number of solutions a quadratic equation possesses.
A quadratic equation has one solution when its discriminant equals zero, Δ = 0. This condition arises due to the properties of the square root function. When the discriminant is zero, the square root of zero is zero, resulting in only one value for x. Mathematically, this implies that the quadratic equation has one real root.
To understand this concept graphically, let’s visualize a quadratic equation on a coordinate plane. The graph of a quadratic equation is a parabola. When the discriminant is zero, the parabola touches the x-axis at exactly one point, indicating the existence of a single solution. In this case, the parabola is said to be tangent to the x-axis.
But what does it mean in practical terms? One real root of a quadratic equation implies that it intersects the x-axis at a single point. In real-life applications, this can represent scenarios where there is only one solution or outcome. For instance, when calculating the time it takes an object to hit the ground after being thrown upwards, a quadratic equation with one solution arises. The object reaches its highest point and then descends, hitting the ground at a specific time.
Q: Can a quadratic equation have no solution?
A: Yes, a quadratic equation can have no solution. This occurs when the discriminant is negative, Δ < 0. In this case, the square root of a negative number is undefined, resulting in no real solutions. However, there may be complex solutions involving imaginary numbers. Q: How can I calculate the discriminant of a quadratic equation?
A: The discriminant can be calculated using the formula Δ = b^2 – 4ac, where a, b, and c are the coefficients of the quadratic equation.
Q: Can a quadratic equation have two solutions?
A: Yes, a quadratic equation can have two distinct solutions. This occurs when the discriminant is positive, Δ > 0. In such cases, the quadratic equation intersects the x-axis at two distinct points, resulting in two real solutions.
Q: What happens when the discriminant is a perfect square?
A: When the discriminant is a perfect square, such as Δ = 4, 9, 16, etc., the quadratic equation has rational solutions. These solutions can be expressed as exact values without any radicals.
Q: Are there any other factors that affect the number of solutions in a quadratic equation?
A: No, the discriminant is the primary determinant of the number of solutions in a quadratic equation. Other factors, such as the coefficients a, b, and c, only affect the specific values of the solutions, not their quantity.
In conclusion, a quadratic equation has one real solution when its discriminant is zero. This occurs when the parabola representing the quadratic equation touches the x-axis at a single point. Understanding the conditions under which a quadratic equation has one solution is essential in solving mathematical problems and real-world applications.