How Many Real Solutions Do Quadratic Equations Have?

When we delve into the fascinating world of quadratic equations, one of the most fundamental questions we ask is: "How many real solutions does this equation have?" Understanding the number of real solutions is crucial because it tells us about the nature of the roots and how the corresponding parabola intersects the x-axis. A quadratic equation, in its standard form, is represented as $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'a' is not zero. The number of real solutions is determined by the discriminant, a powerful part of the quadratic formula, which is calculated as $ oldsymbol{\Delta = b^2 - 4ac} $. This value, the discriminant, acts as a gatekeeper, revealing whether we'll find two distinct real solutions, exactly one real solution (a repeated root), or no real solutions at all (meaning the solutions are complex conjugates). Let's explore how this applies to different quadratic equations, examining each one to determine its unique solution profile. We'll dissect each equation, simplify it into its standard form if necessary, and then apply the discriminant to unveil the number of real solutions each possesses.

Analyzing Your Quadratic Equations

Let's take a closer look at the quadratic equations you've provided and determine the number of real solutions for each. To do this effectively, we'll first simplify each equation into the standard quadratic form $ax^2 + bx + c = 0$. Once in standard form, we can calculate the discriminant ($ oldsymbol{\Delta = b^2 - 4ac} $) to ascertain the number of real solutions.

Equation 1: $y = 12x^2 - 9x + 4$

This equation is already very close to the standard form we need, but since it's presented as $y = ...$, it's technically representing a parabola's graph rather than an equation set to zero for finding roots directly. However, if we interpret this as finding where the parabola crosses the x-axis, we'd set $y=0$. So, let's consider the equation $12x^2 - 9x + 4 = 0$. Here, we have $a = 12$, $b = -9$, and $c = 4$.

Now, let's calculate the discriminant:

$ oldsymbol{\Delta = b^2 - 4ac} $

$ oldsymbol{\Delta = (-9)^2 - 4(12)(4)} $

$ oldsymbol{\Delta = 81 - 192} $

$ oldsymbol{\Delta = -111} $

Since the discriminant ($ oldsymbol{\Delta} $) is **negative** ($ -111 < 0 $), this quadratic equation has no real solutions. The parabola represented by $y = 12x^2 - 9x + 4$ does not intersect the x-axis at any point. It either lies entirely above or entirely below it, depending on the sign of 'a'. In this case, since $a=12$ is positive, the parabola opens upwards and its vertex is above the x-axis.

Equation 2: $4y - 7 = 5x^2 - x + 2 + 3y$

This equation involves both 'x' and 'y', and it's not in the standard quadratic form we need for finding solutions in terms of 'x'. To determine the number of real solutions for 'x', we need to rearrange this equation into the standard form $ax^2 + bx + c = 0$. Let's first isolate the 'y' terms on one side:

$ 4y - 3y = 5x^2 - x + 2 + 7 $

$ oldsymbol{y = 5x^2 - x + 9} $

Again, to find the real solutions (the x-intercepts), we set $y = 0$. This gives us the quadratic equation $5x^2 - x + 9 = 0$. In this equation, we have $a = 5$, $b = -1$, and $c = 9$.

Let's compute the discriminant:

$ oldsymbol{\Delta = b^2 - 4ac} $

$ oldsymbol{\Delta = (-1)^2 - 4(5)(9)} $

$ oldsymbol{\Delta = 1 - 180} $

$ oldsymbol{\Delta = -179} $

As the discriminant ($ oldsymbol{\Delta} $) is **negative** ($ -179 < 0 $), this quadratic equation also has no real solutions. The parabola $y = 5x^2 - x + 9$ does not touch or cross the x-axis.

Equation 3: $10x + y = -x^2 + 2$

Similar to the previous equations, this one also contains 'y'. To find the real solutions for 'x', we need to rearrange it into the standard quadratic form $ax^2 + bx + c = 0$. Let's isolate 'y' first:

$ oldsymbol{y = -x^2 - 10x + 2} $

Now, setting $y = 0$ to find the x-intercepts, we get the quadratic equation $-x^2 - 10x + 2 = 0$. For this equation, we have $a = -1$, $b = -10$, and $c = 2$.

Let's calculate the discriminant:

$ oldsymbol{\Delta = b^2 - 4ac} $

$ oldsymbol{\Delta = (-10)^2 - 4(-1)(2)} $

$ oldsymbol{\Delta = 100 - (-8)} $

$ oldsymbol{\Delta = 100 + 8} $

$ oldsymbol{\Delta = 108} $

Because the discriminant ($ oldsymbol{\Delta} $) is **positive** ($ 108 > 0 $), this quadratic equation has two distinct real solutions. This means the parabola $y = -x^2 - 10x + 2$ crosses the x-axis at two different points.

Equation 4: $y = (-x + 4)^2$

This equation is interesting because it's already in a form that clearly shows a squared term. Let's expand it to see the standard quadratic form.

$ y = (-x + 4)(-x + 4) $

$ y = (-x)(-x) + (-x)(4) + (4)(-x) + (4)(4) $

$ y = x^2 - 4x - 4x + 16 $

$ oldsymbol{y = x^2 - 8x + 16} $

To find the real solutions, we set $y = 0$, which gives us the quadratic equation $x^2 - 8x + 16 = 0$. Here, $a = 1$, $b = -8$, and $c = 16$.

Now, let's compute the discriminant:

$ oldsymbol{\Delta = b^2 - 4ac} $

$ oldsymbol{\Delta = (-8)^2 - 4(1)(16)} $

$ oldsymbol{\Delta = 64 - 64} $

$ oldsymbol{\Delta = 0} $

Since the discriminant ($ oldsymbol{\Delta} $) is zero, this quadratic equation has exactly one real solution (also known as a repeated root or a double root). This means the parabola $y = x^2 - 8x + 16$ touches the x-axis at precisely one point, which is the vertex of the parabola.

Summary of Solutions

To recap our findings:

• Equation 1 ($12x^2 - 9x + 4 = 0$): Discriminant = -111. No real solutions. The parabola does not intersect the x-axis.
• Equation 2 ($5x^2 - x + 9 = 0$): Discriminant = -179. No real solutions. The parabola does not intersect the x-axis.
• Equation 3 ($-x^2 - 10x + 2 = 0$): Discriminant = 108. Two distinct real solutions. The parabola intersects the x-axis at two points.
• Equation 4 ($x^2 - 8x + 16 = 0$): Discriminant = 0. Exactly one real solution (a repeated root). The parabola touches the x-axis at its vertex.

Understanding the discriminant is a fundamental skill in algebra that allows us to quickly assess the nature of a quadratic equation's roots without having to solve for them directly. It's a powerful tool for visualizing the relationship between a quadratic function and the x-axis.

For more in-depth information on quadratic equations and their solutions, you can explore resources like Khan Academy. They offer comprehensive lessons and practice problems that can further enhance your understanding.