Quadratic Equations Practice Problems - Solve by 3 Methods

Let's recall the quadratic formula:

We'll substitute the given data into the formula:

$x={{-(-10)\pm\sqrt{-10^2-4\cdot2\cdot(-12)}\over 2\cdot2}}$

Let's simplify and solve the part under the square root:

$x={{10\pm\sqrt{100+96}\over 4}}$

$x={{10\pm\sqrt{196}\over 4}}$

$x={{10\pm14\over 4}}$

Now we'll solve using both methods, once with the addition sign and once with the subtraction sign:

$x={{10+14\over 4}} = {24\over4}=6$

$x={{10-14\over 4}} = {-4\over4}=-1$

We've arrived at the solution: X=6,-1

To determine the coefficient $a$ in the given quadratic equation $y = -x^2 - 3x + 1$, follow these steps:

• Step 1: Recognize the form of the quadratic equation as $y = ax^2 + bx + c$.
• Step 2: Identify the $x^2$ term in the equation $y = -x^2 - 3x + 1$.
• Step 3: Determine the coefficient of the $x^2$ term, which is in front of $x^2$.

In the equation $y = -x^2 - 3x + 1$, the term involving $x^2$ is $-x^2$, where the coefficient $a$ is clearly $-1$.

Hence, the value of $a$ is $a = -1$.

To solve this problem, we'll follow these steps:

• Identify the given quadratic equation.
• Compare the given equation with the standard quadratic form $ax^2 + bx + c$.
• Determine the value of $c$ by direct comparison.

Now, let's work through each step:
Step 1: The given quadratic equation is $y = -5x^2 + 4x - 3$.
Step 2: The standard form of a quadratic equation is $ax^2 + bx + c$. We need to match the coefficients accordingly.
Step 3: By comparing the terms from the equation with the standard form, $a$ is the coefficient of $x^2$, $b$ is the coefficient of $x$, and $c$ is the constant term or the independent number.

Therefore, from the equation $y = -5x^2 + 4x - 3$:

• $a = -5$
• $b = 4$
• $c = -3$

Thus, the value of $c$ in the quadratic equation is $c = -3$.

To identify the coefficients from the quadratic equation $5x^2 + 6x - 8 = 0$, follow these steps:

• Standard Form of Quadratic Equation: Compare $5x^2 + 6x - 8$ with the standard form $ax^2 + bx + c = 0$.
• Identify $a$, $b$, and $c$:
  • $a = 5$: The coefficient of $x^2$.
  • $b = 6$: The coefficient of $x$.
  • $c = -8$: The constant term (independent of $x$).

Therefore, from the equation $5x^2 + 6x - 8 = 0$, the coefficients are identified as $a=5$, $b=6$, and $c=-8$.

Comparing with choices, we find that choice 2 is correct: $a=5$, $b=6$, $c=-8$.

Thus, the coefficients are identified as $a=5$, $b=6$, $c=-8$.

The quadratic equation we have is $x^2 + 4x - 5 = 0$.

We'll compare this with the general form of a quadratic equation: $ax^2 + bx + c = 0$.

1. Identify $a$: The coefficient of $x^2$ in the given equation is $1$. Therefore, $a = 1$.

2. Identify $b$: The coefficient of $x$ in the given equation is $4$. Therefore, $b = 4$.

3. Identify $c$: The constant term in the given equation is $-5$. Therefore, $c = -5$.

Thus, the components of the equation are:

• $a = 1$
• $b = 4$
• $c = -5$

The correct answer to this problem, matching choice id 3, is:

$a=1$ $b=4$ $c=-5$