What are the steps to solve a quadratic equation in standard form?
To solve a quadratic equation in standard form, you can follow these steps:
Step 1: Identify the coefficients
In a quadratic equation in standard form, the general pattern is: ax^2 + bx + c = 0. Identify the values of coefficients a, b, and c.
Step 2: Determine the discriminant
The discriminant (Δ) is calculated using the formula: Δ = b^2 - 4ac. Determine the discriminant using the identified coefficients.
Step 3: Analyze the discriminant
The discriminant tells us how many solutions the quadratic equation has. If Δ > 0, there are two distinct real solutions. If Δ = 0, there is one real solution (a perfect square). If Δ < 0, there are no real solutions (only complex solutions).
Step 4: Calculate the solutions
Depending on the discriminant's value, calculate the solutions using the appropriate formula(s) below:
If Δ > 0, there are two distinct real solutions:
x = (-b + √Δ) / 2a and x = (-b - √Δ) / 2a
If Δ = 0, there is one real solution:
x = -b / 2a
If Δ < 0, there are no real solutions, only complex solutions:
x = (-b ± i√(-Δ)) / 2a, where i is the imaginary unit (√(-1)).
Step 5: Simplify the solutions
Simplify the solutions if possible. If the solutions are irrational or complex, leave them in the appropriate form.
You have now solved the quadratic equation in standard form!
To solve a quadratic equation in standard form, you can follow these steps:
Step 1: Identify the coefficients of the quadratic equation. The standard form of the quadratic equation is ax^2 + bx + c = 0, where a, b, and c are the coefficients.
Step 2: Determine the discriminant. The discriminant is the expression under the square root in the quadratic formula and helps determine the nature of the roots. It is calculated as b^2 - 4ac.
Step 3: Evaluate the discriminant. Depending on the value of the discriminant, the quadratic equation can have different types of roots:
- If the discriminant is positive (greater than zero), then the equation has two distinct real roots.
- If the discriminant is zero, then the equation has a single real root (a perfect square).
- If the discriminant is negative, then the equation has two complex conjugate roots (imaginary roots).
Step 4: Use the quadratic formula to find the solutions:
- If the discriminant is positive, the formula is: x = (-b ± √(b^2 - 4ac)) / (2a)
- If the discriminant is zero, the formula is: x = (-b ± 0) / (2a) = -b / (2a)
- If the discriminant is negative, the formula is: x = (-b ± i√(-D)) / (2a), where i is the imaginary unit (√(-1)).
Step 5: Simplify and evaluate the solutions. If there are two distinct real roots, simplify the expression. If there are complex roots, express them in the form a + bi, where a and b are real numbers.
Step 6: Check your solutions by substituting them back into the original equation. If the solution satisfies the equation, then it is correct.