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x^2 - 4x + 3 = 0
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x^2 - 4x + 3 = 0
x = 1 or x = 3
Quadratic Equation -- 2 real solutions
Factoring
Quadratic Formula
Completing Square
Graphing
Step 1 Identify the equation
x^2 - 4x + 3 = 0
This is a quadratic equation in standard form ax^2 + bx + c = 0, where a=1, b=-4, c=3.
Step 2 Find two numbers that multiply to c and add to b
? x ? = 3 and ? + ? = -4
We need two numbers whose product is 3 (the c value) and whose sum is -4 (the b value).
The numbers are -1 and -3 because (-1) x (-3) = 3 and (-1) + (-3) = -4
Step 3 Write in factored form
(x - 1)(x - 3) = 0
Rewrite the quadratic as a product of two binomials using the numbers found in Step 2.
Step 4 Apply the zero product property
x - 1 = 0 or x - 3 = 0
If the product of two factors equals zero, then at least one of the factors must equal zero.
Step 5 Solve each equation
x = 1 or x = 3
Add 1 to both sides of the first equation, and add 3 to both sides of the second.
x = 1 or x = 3
Verify
Check x=1: (1)^2 - 4(1) + 3 = 1 - 4 + 3 = 0 ✓
Check x=3: (3)^2 - 4(3) + 3 = 9 - 12 + 3 = 0 ✓
Real estate connection: If this equation modeled a property's break-even analysis where x represents years and the equation represents net cash flow, the property breaks even at year 1 and year 3, with negative cash flow between those points (like a renovation dip).
Check x=3: (3)^2 - 4(3) + 3 = 9 - 12 + 3 = 0 ✓
Real estate connection: If this equation modeled a property's break-even analysis where x represents years and the equation represents net cash flow, the property breaks even at year 1 and year 3, with negative cash flow between those points (like a renovation dip).
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