Math Problem Statement
Solve the following quadratic equations using the quadratic formula.
Solution
The image contains six quadratic equations that need to be solved using the quadratic formula. Here's a step-by-step breakdown of each equation:
Quadratic Formula:
x=2a−b±b2−4ac
Where a, b, and c are coefficients from the general quadratic equation ax2+bx+c=0.
Solutions:
-
Equation (a): x2+x−1=0
- Coefficients: a=1, b=1, c=−1
- Using the quadratic formula:
x=2(1)−1±12−4(1)(−1)=2−1±1+4=2−1±5
Solutions:
x=2−1+5,x=2−1−5
-
Equation (b): 2x2=5x+3
Rearranging:
2x2−5x−3=0
- Coefficients: a=2, b=−5, c=−3
- Applying the quadratic formula:
x=2(2)−(−5)±(−5)2−4(2)(−3)=45±25+24=45±49
Solutions:
x=45+7=3,x=45−7=−21
-
Equation (c): 2x2+x−5=0
- Coefficients: a=2, b=1, c=−5
- Applying the quadratic formula:
x=2(2)−1±12−4(2)(−5)=4−1±1+40=4−1±41
Solutions:
x=4−1+41,x=4−1−41
-
Equation (d): x2+7x−1=0
- Coefficients: a=1, b=7, c=−1
- Applying the quadratic formula:
x=2(1)−7±72−4(1)(−1)=2−7±49+4=2−7±53
Solutions:
x=2−7+53,x=2−7−53
-
Equation (e): (x+1)(x−5)=−3
Expanding:
x2−5x+x−5=−3⇒x2−4x−5=−3
Rearranging:
x2−4x−2=0
- Coefficients: a=1, b=−4, c=−2
- Applying the quadratic formula:
x=2(1)−(−4)±(−4)2−4(1)(−2)=24±16+8=24±24=24±26
Solutions:
x=2+6,x=2−6
-
Equation (f): 3x2=4x
Rearranging:
3x2−4x=0⇒x(3x−4)=0
Solutions:
x=0,3x−4=0⇒x=34
Would you like more details on any specific solution?
Related Questions:
- How do you derive the quadratic formula?
- What are the discriminant's roles in quadratic equations?
- How do you solve quadratic equations by factoring?
- Can the quadratic formula give complex solutions?
- When is completing the square useful in solving quadratic equations?
Tip:
Always check the discriminant (b2−4ac) before solving to determine the nature of the roots (real or complex).