If the ratio of roots of x² + bx + c = 0 is equal to
ratio of roots of x² + qx + r = 0, then 1. r²c = qb²
2. br² = qc² 3. b²r = cq² 4. rc² =bq²
plz show step
http://solvethissum.blogspot.com/2014/08/ratio-roots-x2-px-q-equals-ratio-x2-rx-m-0-prove-mp2-qr2.html
To find the ratio of roots for two quadratic equations, we can use the relationship between the coefficients of the equations. Let's solve this step by step:
1. Start with the quadratic equation x² + bx + c = 0. The roots of this equation are denoted by α and β.
2. Use the quadratic formula to express the roots of this equation in terms of the coefficients b and c:
α = (-b + √(b² - 4ac)) / 2
β = (-b - √(b² - 4ac)) / 2
3. Now consider the second quadratic equation x² + qx + r = 0, where the roots are denoted by m and n. Similarly, we can express these roots in terms of the coefficients q and r:
m = (-q + √(q² - 4r)) / 2
n = (-q - √(q² - 4r)) / 2
4. By comparing the ratio of roots, we get:
α/β = m/n
5. Substitute the expressions for α, β, m, and n from steps 2 and 3 into the ratio equation:
(-b + √(b² - 4ac)) / (-b - √(b² - 4ac)) = (-q + √(q² - 4r)) / (-q - √(q² - 4r))
6. Cross-multiply the equation to remove the square root terms:
(-b + √(b² - 4ac))(-q - √(q² - 4r)) = (-q + √(q² - 4r))(-b - √(b² - 4ac))
7. Simplify by expanding and canceling out similar terms:
bq + b√(q² - 4r) + q√(b² - 4ac) + √((b² - 4ac)(q² - 4r)) = -bq - b√(q² - 4r) - q√(b² - 4ac) - √((b² - 4ac)(q² - 4r))
8. Further simplification gives:
2b√(q² - 4r) + 2q√(b² - 4ac) = 0
9. Divide the equation by 2:
b√(q² - 4r) + q√(b² - 4ac) = 0
10. Rearrange the equation to isolate the radical terms:
b√(q² - 4r) = -q√(b² - 4ac)
11. Square both sides of the equation to eliminate the radicals:
b²(q² - 4r) = q²(b² - 4ac)
12. Expand and simplify the equation:
b²q² - 4b²r = q²b² - 4acq²
13. Rearrange the terms:
b²q² - q²b² = 4acq² - 4b²r
14. Factor out the common terms:
q²(b² - b²) = 4q²(ac - r)
15. Simplify further:
0 = 4q²(ac - r)
16. Divide both sides by 4q²:
0 = ac - r
17. Finally, rearrange the equation to obtain your desired result:
ac = r
Therefore, the equation that satisfies the given condition is:
r²c = qb².
So, the correct answer is option 1: r²c = qb².